How To Make a Cycle Analysis - Correspondence Course for the Foundation For The Study of Cycles

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Page 2

Leeeon VII

-

Bow to treke and use the perlodtc table to reveaL the typlcal or average ahape, etrength, and tlming of tbe eycle. Eon to rotate the perlodllc table to coupeneate for trend. Uee of color lu the perlodlc table.

Leeaon VIII

-

Esu to use a pertodlc cycIe.

Leseon IJ(

-

Eow to poettlon

Leseon X

-

Randouls: Three vayg to nlnlmtze

Leseon XI

-

The effeeta of movlng aver€geg upon trendr upon randon numbers.

Leseon J(II

-

Devlatlone of nrrmbere fron uorrlng averageg of vartous Iengths. Eon to correct for dletortloa.

Leseon XIII

Lesson XIV

table

to deternrlne the lengtb

of the

the cycle. then. upoD cyclee,

and

Three lrays to cleflnltlze the eycle. Deflnltlzlng cycJ,es of lntegral length antl of fractlongl Deternlnlng the lengtb. calendar tlntng of a cycle. -

Leseon XV

Hotr to nahe a perlodogratr. Slne and coslne curves and bow to flt

them.

Lessoo XVI

-

How to nake a sluple haruonlc analysls. Bm to nake a Uee and llnltattone of tratmontc mu1tlple harnontc analysle. an8lys1e.

Lesson lffII

o

How to courpute movlng Bercentages, novlDg rat!.oe, and novlnc tllfferencee. Thelr effect ugon cyclea. Eor to use thesl to detect hldclen cycles.

Lesaon XVIII

-

The etratght llne trencl. t{hen to uge lt A ehort cut nethorl for cotrputtDg tt.

teeson XIX

-

Eov to uge p€rtodle enother,

Leeson Xl(

-

Wetghted novlng averages. cycles.

Legsou XKI

-

Eotr to mke a Strelff

Leseon }C(II

o

Eor to deterulue the future.

Leseon XXIII

-

Teste for Part II

tablee

cycle ana1ysls.

to separate one cycle froo

Eow to use tbem to reveal

btclden

Analysla.

trend.

signlflcance

for

Esv to proJect

trcnd aud eyclee lnto

of cyelee.

constets of about tOO pages,

In ecldlltlou the IesBoBs lnelude, at approprlate standard practlce tnstructlone and eupplenental by the author ln tbe conduct of bts ot'n work.

placea, naterlal

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Contents Page 2 LESSONIV ST'PPI,EMEI.ITS: The Glossary The Log Conputlng Per Cent of Trend

1. 2. 3.

TESSONV - CYELEAI{ALISIS Eow To Make a Cycle Analysle Nunbers - An Outllne A Detalled outltne S o m eB a e l c C o n c e P t s

A. B. C.

of A Sertee of

LESSONV SUPPI.EMET{IE Conblnlng CYcles Reverelng CYeIee CYeIes ln ComnodltY Futureg Cyele AnPlltudee About Problen lr

l. 2. 3. l. ,.

I,ESSONVI - RO}I 1'O CET BII{TS OF CICT.ES gCH TO MAKEAND USE TIIE TIME CHART STTUjUIARD PRACTICE INSTRUCTIONS

5 ' Indexes - 5 - The Gracluatctl Scale -7-TlmeCharts

: I.ESSONVI ST'PPLEMENTS I. 2. 3. lt. 5. 6. 7. 8. g. IO. Il. !2. 13. TESSONVII

A Short cut or Bastarcl Method of t'laklng A Tlne Chart Tlne Charte Tlure Charta, Contlnued Tlme Charta, Contlnuecl The Tlme Chart Problens Solutlon to Problena Solutlon to Problen 3 Solutlon to Problen l, Contlnued lrlaklng Tlare Charte l{ork Harder Another Tlue Chart More About How To gat Elnta of Cycles The Eosklns Ttne Chart CongtruetlnS. The Tlne Chart

- EOJ TO I.TAKEAND USE A PERIODIC TAETE - PATI 1

STANDANDPRACTICEII{STNUCTTONS- 8 . PCTIO(IIC TAbICA

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Contents Page I+ LESSONXII

- A - SUPPLEIIEiIT I - Reconstruetlng A Cyele Fron Its ltloving Average

LESSONXIII

- DEFINIIIZ]NG THE CYCLE

LESSONXIII

: SUPPLEI'IENTS I. 2,

More About DeflnltLztng Cyeles The Deflnltlzed 9.l5-Year Cyele In The Standard & Poor's Corporatton CornblneclIndex - A - REVrE\I, LgSSoN I - LESSoNXIII

IASSONXIIr

THE PENIODOGRAI'I

LESSONXIV

L LESSONXIV SUPPLEMEITT

More About Perlodograns

I,ESSON)ff - SfNE-COSINDCURVESAND H0lit T0 FIT TIIEI'! I,ESSONXV SI'PPLE},fENTS: I. 2. 3.

Brlef Table of Slnes and Coslnes A Table of Sines and Coelnes of Anglee Expressed As Percentages of Perigons Flttiag Sine-Cosine Curves

ANALTSIS TESSONXVI - HOIJTO I."AKDA SIMPI,E RARMONIC LESSON)[VII

- MOvIllc PERCENTAGES, ["!OVINGRATIOS, Al{D DIFFERB}ICES I','IOVING

: LESSONXVII ST'PPLEI,IEIITS 1. 2. 3. l. LESSON)$III

Movlng Differenees Irtoving Percenteges, Movtng Ratlosr aud Movlng Dl fferences, Contlnued Questions About Movlng Percentages and Dlffereucee The Effect of Movlng Dlfferences - THE STRAIGI{TLINE TNEND. }IEEN.TO USE ONE . ONE HOI.ITO COI,IPUTE

LESSONXrX - TIIE PEnroDrC fABLn, coNclt DED I,ESSON)O( . WEIGTIIEDMOVINGAVERAGES : I,ESSON)O( SUPPLEIIII1IJTS 1. 2.

W , C ' Y e a t r o r n t sP a P e r Yeatnen:s AnnJlficatton

Table

LESSONXXI - TIIE STRIITF I'ETHODOF CYCI,EAMLYSIS I - Short Cut for IESSONXXI SUPPLEI\ENT

Stretff

Analysts

I,ESSON]OUI - HOWTO DETNE:I]'II ATIDPRq]ECT TREND

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Introductton Page 2 r (1) (Z) 0, (4)

Conquting nachlnes (vhlch add, substract, utrltlpty, Sltde rrrles (trhlch umltlply and c[lvlde). Attdlng nachlnes (vhich ad.d.and, subt ract ). Iogarithmg.

and dlvlde) '

Iogarlthns help because they enable you to substitute addition anrl aubtracantl dlvislonr ond to eubstltute divlelon for the etrtractton for nmltlpllcation tion of roote. ( Oo not let the vord logarithar scare you. Logaritlue &re as €esy es pte. Operating a tou can loarn all yor need to hlou nbout then ln flfteen nlmrtes. gllde rule le €EBy. too.) If you do not have aceess to a conlnrtlng nachlne yotr ean get along sstlsYou v111 aot Iou rr111 need. one or the other' vlth a slld.e rule. factorily aeed both. srachine you Day wlsh to r eat or If you do not have acc€ss to a ealculatlng The i{onroe, Fricycle analysis. for borrov one. Conptoneterg are not suitable you not flntl all wtl1 but good. nachtnes, d.en, and. l.larcrrant courparrles all nake 9rlden. ig the of nake preference lly nod,els eultable for your vork. one of the I nyself $otrltt \y Rather than spencl noney on nachlne rental, hand. [hoy by opereted uhlc! are l{onroes aesk nodel so-called &ecutlve Itttle as little for as $60' can often be bou.Sht eecond'hand rul'e But, aB eald above, you do not need. a ealculatlng nactrlae. A sllde logarttlng anct rule sllde A veII dlvtcling and d.o your rmrltlplylng ulll .enough' tlllld'oyourroots'slrdarraddingnachine(oryourheedt)rrllldoyouradltlltlon and. subtractton. gllde tttle' If you d,o not have accese to a conlnrtfn€ nachlne you gggg have a any Sears If you do not have one already, Vou can.bW i pretty good one fron Instnretl'ons for Boebuck t: Conpany natl order iroo." for $3 .?5, pfut postage' gllde nrle' 0r, lf you trreh, r rrllr rena you a etudent operatl'g come gibh lt. Send ;2.00 depoalt, returnable at any tlme' get alorrg vlthout If you ere pretty good at adding arrd subtracttng you ca4 you nay Houever, a nachine rvill save lrou so tnrch tlme that an addlng nachlue. utrch how you 8ee vant to rent or borrov one. you can decide thls later trhen vork le lnvolved and borr ruch t ine it takes you. get one rrith a subtraction If you clo rent a^n adaling; nachlne, be sure to to tbe prrchaee prlce' be applieil can key, And nake a deal 8o that the rental lf , lst er, Yor declcle to buY.

tha-rtlne you trlll need for the cotlee' I r.rill eulply.you rrlth vhatwer clrart paper you ean brly it direct fron the If you vant nore paper for your ot'rn puqposes I asr asking thls conpa4r to sead GocrexBook comps,ny, Nor*ood, uassachu""tt". you rrirl ie surprtsed to see ho'tr&Dy klnds of paper are you a eatarogue. ava!IabIe.

'soTra8 a1c.tc 'lcsJ uI (.e1sf,1aue BeTJoBeETl p€Ttec saurleuos sT s1s.d1ew ',saTJes euTl TTa aJ€ ' ',ale ogrl qlTA prguJef,oc erB sas.{1eua e1c.,tc TI€ {TT€cTlc€Jd ."p1"i.C do.rc .C1.rae.c'sa.rn91g: uoTtcnpord ^d1q+uou 'sac1.rd, {co18 {TT8p 'ee-mle.radue1 aql 'etrTl Jo rePro uT .C1.rnoqaATBBaconS 'saIJaE suTl € paITBc sT BaTras ls€llnser ,uolsseccns Jaqlo .firpu rraqlo pe?rerrs e.rtl sraqurur aql uaWl u1 JelJp euo aql .'s.(bR '8.reqtrnu Jo rrT puB .en11. gio repJo uT .az1s Jo Japro u1 pa?ur.r,re eq uBc sJeqtrq[ sproA aq&) rr€ sT sraqmu Jo saTras V '1EEE6?'[I€E- slr€otu 66lfff TAEft$ffi

'€,567 --#gt

,satsraas la{rBu ttcols Jo 8aTras € pqTJ TITII nO,t pesolcus leeq8 € u0

'sa.m91g:le{r€[ )tco?s saos Jo als.d1are e1c^dce Fu1:Jau,tq, asrnoc eql 1rla18 aJoJaraql ITIA erl

srousTsi'up11 eqrpeldops a^cqr 'tuT+saraluT ;i-iilT"'lufirtl.;i::$'.:H::;

'ue1ag pa{oTJqB .ls€eq'notr reu g:o og r.Jelq?f l er{l TIB raq pTeq trctr .C1;nrqaq1 lnq t 's8u1q1 .;ro eTppTq aql uT lrBls ol 8TtaTr ., '&esgacoq E€ {mJ1:lc€ q ol uar{fi .ool .as.moo ecuepuodee&roce JoJ aq pInoqs +T pu€-EoTrots roJ deH ulapou eqf 'BT XTnf uo 'pue1?ng 'aulqspo suag ' ' . rten 8 s .C1eue.r1:o l 1 ,, 5gg1 Tq6 TTnp 'uoTt{s€; ltl.rapao uT s8u1q1 fnd oq't sI l€ trJoq BTn raqle3rpuu.r8 s ruoouTqog ualag,r .(?rr peuoTqs€J pTo eqfi ,{ooq lxel € se-J.1e1s B elTJr.r oq s.dbn onl oJre erar{J,

EEffi'qT

F E€G

'asJnoc aqt uT retaT pereaoc eq o 1 'os 'nod JI ITaq ITTn r ?u1o? erc ls{tl stuTr{t of uT€l.rad .t\rd suoTlsanb .rno.( g:o auos pug .ao.rnoo eg," ltq paleaoc ?grt egl apTslno suol4senb Jensu€ ol eryatrepun louu€c 1 'es.rnoc JO 'pufqaq 'ratlaq a{l eJou srt[ 'luerl no.( 1T€ lJeT {sV 'pIno/l esTr.rJaqlo nof s€qt esJnoc tuTeq rcrJ no.Cdsa{ ol clTer{TITA suoTlsar$ eql Jo lno arotr 1a? o3 no.Cdleq TITrr suoT?sanb 'u1efu lT a^Tt I uaqn rellaq esrrtoo eql alFq ol au dleq ITTr{ suoT?sanl 'sn }o qloq. dleq ITTr.l suoTlsarti" 'puelsraptrn lou op no.f, 8o1r{t -f,u? sT eJaql JT suollsanb {B€ oS 'puTqaq nol( eatrdl pue uo of rfuu I 'dleq paeu no.f uaqA Xno rao.uauar puy '1.red .rnof, op noA g:1 'nof aos l€rll lou louu@ TT€c I 'r1od +BT8s€ uo )tJon lnor{+TA 1Tta[ns aql o1 nod rt.r.rec ol nol I]tB € lou t;rtsI lnfi '{ror.r aql op lsnu nof, 'uTelrmou e SulqrrTlc 'epTnt sa uoTlcunJ I1TA I IITI{ I 'sTE a{TT 'lTnsa.r sTr{l fuTaaTr[cc .roglXlTTTqTsuodsa.ra{t a{Bl lsnru dd 'relaRog -{Teup elcfc at1erl o1 noA qr€a I ol sT--ouTtrl pu€ s.rrpI--oaTlcaf qo T€ntnu JnO

EroTiE?-r6 'uaql AoIToJ ostseTd .sa.rlnbeJ uoTs€cco se nort o1 ual13 eq ITTI\ suoTlcn rlsuT aoTlce.rd pJ€p -Irefg 'pqa aqt u1 Xod EacTlce:d p{€prmls urB?Jec lql ur.roqs s€q acual.rad:q6

6T@f

T'@,G-+'S

r 6 aSed uoTlcnpoJlur

Introduetlon Page 4 r

Iou vl11 proceed as follovs: The flrst

thing you vill

do rl111 be to chart the data.

(gatp arc Durnerlcal faets. The vord. ts plrrral. (tUe aingular, rarely used, nThe ls d.atrur.) Iou nwer say data !g.. . .rr lou ahlaye say "th" data sIS . ' . rr !!he vord data ls pronounced. d,ayr-tahi not alaht-tah, or datl-ah., fou w111 clrert the ilata for (f )

You want to see lf

hro reasons:

the series shous a @!.

(Trend. ls the tendency of a serles of flgrres gradually over a period of tlne. )

to lncrease or decroasc

Q ) Torr vant to dlscover the atrrproxinate Length of the cycleg yor rant study flrst.

to

(Qfgf_gg are osclllatlons--ups arrd d.orvns. The sord eones fron a Oreek vord neanlng clrclea eonlng aroundl agsin to the place of beglnuing, e.!., a low. fou y111 nant to etudy theee up antt d.ovn aotlong ln the hope tbat tbey trlIl be rtlythnic. ) (nfrvtUr mea.asto npeat vith neaeured tlne. )

a beat,

It

cones from a Greek word oeaning

heve (Ot cgtrrse, .you aro hopeful that the rtryttm wlll be so regular ancl vill of result be the cannot reasonably repeated so nary tlnes that itre re$rlarity But this gets lnto tbe cbanee. In this event you can e:rpect it to contlmre. aatter Of waluatloa, trhich trill l>e conslderecl later. ) If, in a:ry lnstanee, the rlata you ar6 stutLylng bave no trend' at once to lnrnt for cYcles,

yotl can Proceed

a 0n the other bancl tf, ag utth tbe stoclc oarket figures, the data d'o show wlth analJrsts' tbe proceed you can trend, xou have to reruove the trend before Iotr usually

remove 1t as follovs:

Flrst, vol flnd vhat the trend values are for . thuetlc arl )

each y€ar ( a uatter

of sluple

value for tbet Secontl, you flnd for each year the percent th'rt the actual of conlnrtlag natter the mereLy Is year ls of the trend varue for lhat year'(turs e f a percenta€es).

for

Iou nou eearch these values ad-Justect for cycles-

tpsnfl-these

percentage of trend-

')t:or.\ ol

tlro11 1a3 ei.t eso .b L6.2

2r.3 3 3 .O 35.2 33.2

3r.2 3r.2 39.9 trO.B 35.It 39.2 b2,0 lo,l 38.2

3?.r 33.9 26.2 28,2 3l{.\ h3.5 ,2.3 lr9.3 l*?.o 39'5 38.3 \h.B b5.1

year

rBSB r88g r89o rBgr rBga rBg: I89lr

rBgl rB95 r89? rBgB IB99 19oO

rgol Lgoz 1903 l90lr

rgoS r9o5 1907 190B 1909

].gr0 rgrr I9r2 t913 rgflr 19]5 t9t6 191?

rgr8 T9I9 I92o I92r

I ndex

\3.b l{lf . h lrl+.o L2.0

\6.3 39.8 36.5 3?.8 3 5 .b 3 7. 2 \2.2 ,2.6 51.3

6r.,

69.9 60.2 5 8 .B T5.L Bo.6

6r.6 6r.o 81.2 78.2 1 7. 3 79.7 ? 1 .r

6 t. \

59.9 Bo.h 72.L 5 l r. I

7\.6 67.8 58.3

Year

Index

L922 Lgz3

7L.5 72.9 76.9 gb.8 105.5 l.2\.9 158.3 2oo.g 158.2 9 9- 5 51.2 5 T .o 76.6 82.g ]I7.5 r1?.5 B S. 2 9\ .2 8 3 .I Bo.o 59'l*

L92t1

Lg25

rye6

rg21 tgzB Lgzg 1930 193r rg32 r933 Ig3h 1935 L%5 r93? I93B L939 rg\o rghr lgL2 1 9t 3 lgtr\ L9\' L)\6 1911? B 191{ 't-orr9 195o 195I

:)r2 L9r3 19r\ L955 L955

9r.9 99-8

rn-.,

L39.9 r23.0 rz\ .l+ .t21.li rlr6.lt

r7s,, 1 8 77 . I-89, o ?26.7

3CO.O

3\l ,5

These data are derivetl from three series vhlch have been spltceil and p'.lt o n a 1935-39 = LOObase to match currently

published clata'

Sources: f85tr-18?O: Clenent Burgess Index.

& Poor's Comblned Index of CommonStock Prtces, L926 = lOO. r9r8-1953:Also Standard & Poorr s Combined

rBTr-r9r?: Standsrd

Index, but f935-39 = 10O. Current monthlY data 8re publtshed 1n the 9!II9I of gfgeq! B u s l n e s s .

'nod ol dn 8T 1I .tutuultaq, aqt uroJJ srapToJ qcns gsn ol qsrr'\.deurnort '.tO 'BaTTJ T€cTlJaa uI pegn lJoB egl Jo srap1oJ ecuapuodee.r.rocol Jo BJepurq raqqo ol leTral€s Jaqlo eql ps€ ssoeset eqq IsJsu€Jt ol l{sTr.t.rteuno{ 'dn eIITJ Jepurq aql s€ 'Ialql '8q€1 xapuT plot{ Pu€ 6uoTlca8 eql ele.redes o+ .raded r{aeaq Jo preoqprttc uTql agn u€c no.{ 'rapuTq, € esn no,t gi1

.tl

suoTlcnJlsuI eaT?c€rd pJ€ptrBls

.9

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.g

sa 11ddng

'T

'e 'p 'o .q .€ 's

Z

.

s quatm€T6sy prrc T€TratEl t uo sseT

.

graTqord

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:snoTIOJ s€--sJopToJ .nro{ cin las Jo-Japurq

'v

€,

-rnorf aPT^Tpqns 'qlTn 1J31e of

'r{sTA nod 311 3.{alr paqctmd nort ol lalralq! os aq .tpaer oITJ 01 sTr{l lT lur{t IT€ Puag TTT^I ',{EA Jor{lo oruos ur ro 's-raploJ aITJ .strapuTq Eu1.r-oar{l pueuuocar I Ilsr{s I tr.rsu1p.rouT '.rapu1q 8u1.r-aerql e uT s:eded {Jo/.r.mo.d oITJ o? t{BTrdrfcu ro1

tutrTd'r SltOI&COU$SilIirClfiCtltcl (IWGi{V,IS

T--rds

Introduetlon Supplenent I

INTRODUCTION Supplenent I CAICUI.ATINGMACHINES The followlng

qucstion,

and ny answer, nay be of general lnterest.

QITESTIO}I: I an wonclerlng why you prefer the Frtden, vhat you thtnk of the llarchant, whether you are farnJ.Iiar with the new models of the Marcbant, and whether a 1O key or I key maehlne 1s needed? AI{SWER: I thlnk

tbe Msrchant 1s a goocl machine,

I am not fanl]lar

with the newest rroilels of any of the nachlnes.

As far as I know an I key naehlne woulcl be adequate for analysis, We have B anC 1O key Monroes end I0 liey Frldens. My preJucllce ln favor of the Friclen ls frankly on the f ollovrlng: 1. A repairman engagecl excluslvely vas the best machlne on the rnarket. 2. Fridens,

ln repairing

any work 1n cycle

a preJucllce.

It

ls baseil

Monroes tolcl ne tbe Frlclen

The head of an accounting departnent of a large company uhich used Monroes, ancl l.tarchants told me the Frlden was the best of the 3 machlDse.

3. Dr. Leonard Wlng, of the Founclatlon Staff, investtgated Monroes, Frlclenst and Marcbants a yeor ago and decidecl in favor of the Friden. Ee said it vas faster than the Marchant. b. e as l } y .

Conparlng Frlden and l'lonroe, I flnd

the Frlden gets out of oriler }ees

lucky 1n prlce on Frld€DBo For exaqpler the nachlne 5. I have been fairly ve bought for Dr. lllng ( second hand, of course) cost only $:a> for a 10 bank nachlne vtth fully automatlc divlslon, nultipllcation, negative nultlpllcatlon, etc. etc.

tt

I admlt that the above le not anough to go ono ls worth.

I pase lt

on nerely

for shat

tAIX uosEa-I alatlatluB oJB sqersopoTrad ararlA .qlaral

qlBuaT aTqeIaP Fru'Jc ls t olgg\l+ suarSopoT.r,ad01 sacualaJau

aurBsaql Jo salc^o .[q pazlJalcBlBr{]

AIB XaPUT

B se}8I?snTTT uB JO sluauoduoc aql usqA uaaa asTJe ,(Bu qJTI{A ,ttTnclJJTp Jaq+JnJ ,glr

aged tt66t

fe6 .ro3rs-ffi

'a1c11.r9 Puocas aq& uoJJ aEed a1Eule B

'g8-eg sa8ed rt66t qcrBl{ aol ffi lualaJJlp

sorJ uaIB+ se111I

'saull

luaJaJJTP 18 sluauoduroc ,(.r1snpuT

.traa 3:o dn apeu aJB I{1'TATBaP aA saoT;d 1eo1s Jo saxapuT aqX 18ql PqB

satcrtc luaraJJTp aABq {co+s Jo ElJos luaraJJTp 1eq1 no^( sAoqs ,,'1aryBn l?ols aqx uT sa1c./t3,,paTTBe ,lsrlJ

aqf

.ffi

u1 paraaddE qf,TqA saTcTue on1 nof

':{JBT slT Aq pacnporluT saTxTxelduoc aql JO

roJ ?uTcnpo.rdar trB I Pua sTql oI

spalncas eq s'tea1e louueo '(11aua8ouo[ OIB1B aq gBc no,( 1sea1 +B Xnq. .saqo JaTTJBa JoJ sarn!1;

aq1 qlTA snoaualggoq aq 1oU '[a

JaIBT JoJ saJnETJ eq+ eaTuedlxocTBnpIATpuTJoJ uaAg JO sar;s

?Ttrc f11::ssrDa:

rs:ea.f, f1.rea aql IoJ

s'rea'[

'palgaJc 1a'( lou salueduoc

'tTTlt f.rlsnpul

uB Jo uoTlcnpo'rd

.X?puI aql Jo s.rea't ^,t1:.laoql JOJ acualslxa uT uaae lou Jo rapul ue,tlrBTTUTS juo11r1aE puB uorsTAaTal sE qcns saTrxsnpuT Jo sorr.31J uTBxuoc 'auTl luasard aql lB

.TTIlt daury1gfo po;rad a:qErapisuoc lCuerano 8u;pualxa

xapuT uB .oldUexa lod

(:saulsnq

Tslaual ;o

'elc? snoaualouoq a^eq 01 sTqTsso'I stenge lou 8T 1I

.3TqTs8Od sB snoanvEouoq sB aq 4Bp lnort lBq} 1ue1ro([tr1 I rll

v&vo J0 r.llglxgconou Z luauaTddns

liormnqcuJNr f ain3 Z +u€-,ra..lrJr.g soll3npurluI

Introduetlon Supplorent 2

OlI IIIT'ROIIICTI Supnl ement 2 Co]'DUCT0F II{]i COURS]I In e"grver to the questlon, nDo your'ant the probleqrs trariler? trlrgler? lbout the aanc?rrone of our nore thouglrtful students trrltes, nlt ls hard for ev€n an actlve Yet by taklng one €aBy step after a,uother, each nan to seale an etgbt foot ualI. ulthln the llmlts of hls ablltty, even an ordlnary person cer cllnb a notrntaltt. It ls the Job of the lnstrlctor to e:rplaln every step go tbat the stud,ent can cover the gror:nd, snoothly and easiJy.rl I belleve this nan speaks for rnost of you andl I sheU be governed I shnll try to take you by the trand. antl lead you snoothly ana aceortlingty. careful-ly to the top of the mountain. Hotrerver, I voulcl nuchrather d.o it the other vrqy. I uould. rather nalce you I vouLd. rather you entled,the courge, skin your shins gettin,l over B-foot ualls. lrit ltke to glve you not tourlsts bur eonnanclos. Itd. like to nake you towht problems that vou1d. reogire you to th:ln!--problems that you night not be abl,e to solve. ll li.ke to make you learn the hard. tr4v. But I clonft thlnk you traat this nncl I shall borr to rrhat I think are your ttlshes. beyontl Hovenrer, lf ]'ou do cone up a,3ainst somoproblem that goes a llttle yor See lf d.ellberately. the text, rr:nernberthat it na..rhale been introduced. (fut for help!) you be to call tf caqlt, eure canlt flgtrre lt out for yourself. I call Gas115' Also, keep me posted es to your reactlon to vhat I an givlng you' aodify the material .end the problens to neet the need.e and tlesires of the naJorlty, so please be vocal. Ancl ask questlons !

g8 'sa8ed Sur,nolloJ oal aql uo suurtoporred aql lpnls no{ yr a?ueraJJIp ;o seldunxa Jaqlo {ueurJaaorsrp uBO notr 'l uel.rodur aroul sI sq?uov LZ;o alcAc atcra,\s ua slro?S 8ur ur 1r1 6C !o al c^c ade P I aC ul pvsql uour -Ja^B uB uBrll luol.rodrrrr ssal qrru Alsnore -1o sr rI alclc qtuour-gt B sr arar{l Jl s{Jo?S tuaudTnbg urBJ ur .aldruexa .ro3 'salcAc

' s 1 c o 1 g S u r u I r t p I o J U B I p B U B J. s 1 c o 1 g IBrJ4Brl turplrnl :s,rroIIoI se s)lcqs tl(lll|ll)3 J o s d n o r i t u a r a f l r p x r s I o s e s d l s u BJ o s1[nser arl] tuasa.rda.r sun.riopor.rad aq1

'sJequrel Jno 'sexa1 'oruoluy uBS 'O Jo auo Jo lcocldo3 'S 'g Aq pareda-rd sa9eraae dnort snouua ' q I J s a r l 1r , i a r a q Jo sue.rtopor:ad xIs I

arnpo.rdar r rce; sr'r1,Hrr;rTTiilrw

luaraJlTp e^Bq slco?s lusraJtIp 18r11 ar aJuBcII1utTs sTrtI .sesodrnd ?uasard Jno roJ eouacTJlrrtTs Jraql la3 o"l ropJo ur IIInJ untlt pustsr?prm ot paau ,{11ea.rtou op no,( fn{ '1.roder sTql }o uortca5 lBf,ruqrrl eqr o? J" f er 's ura. rS oporrads, poddo3 ' Jl t 8 u r 1 a r [ r a t u l u I d l e q e J o u tt u a , i n o , { ; 1 ' {l t ua 1 uI sq?uou 9g Jo aa3r lafuo.rts uaAe uB s r oreql ' e8B r3A B or ll uo 'l B g1 sbl ecl pul sq?uour9t re luad r3q?ouB sarJas eures arl? uI '{ 18ual s }qt o? eso[c { .r a,r ro 18 aA B ' i u B s a t B s T p u l 8 q 1 u o u r0 t t B l e a d rlrrrer eql s{ctrls luedun3 luaunsaAul paSeraeal 'alduexa oJ J Jo ua.rtopot.radoql ur 'os[B alcAc aart llada: Jo .luqll{r B eq ,{eua r .raqralclc rlrlrer tus?-rodur ue st eJar{l alaqa aaJnof, Jo puy 'aJuol.rodur ue pazdleps 3:oalolc r!rltrr uaeq e^Bq sa.rn8t5 aqt ur ar araql 1 n q l ',ttlq sr un.rSoporrad I Jo earnc aqt ararlr qr8ua1 a1c{c I-rare rBau ro 1B 'Ilqtnor flurleadg 'lIa[ eqt tB elBcs aql Aq ttaotls sr alclc eqr Io acus?rodur a^t?Blar aql 'rlr .r Sopor. rad aq? Jo uro?t oq eqt 1B al ers eql uo tur oqs aJB s qtS ual alc lc l uaraJJrp a{L '8alr€lI Io sarJes B u-r tuesa.rd salc{c aflnrarr snor.raa Jo ecuEl.rodun eAr?B[eJ eq? Surnorls e {1a.ranrsr up.rtoporrad y lrBrlc '0S6I ;o {uq o'r (fiil {renmf utorJ 'BFco?gtuaudTnbg urBJ pue 's1co1g IIV pac tJ d aql s no"us ll ecs rl ,l l '"r fc o?S eJntf rd rpnon 'slcqts ftndunn tuauqsaaul patnraral

tBqnaups 18 prrB 'stq?ran tuaraJJIp tBqaetDs r{r!r salclc ?uros eq? ro ,salcdc luerat -JIp af,uaprAa s{ro?s Jo sdnori ro slco?s tuareI.I!p roql lrBI aq? sr {ioloqcdsd ssBru sal ns sau s a q ? ? a { J e l u ? B r l l r i l er A }- o I c o l tutod eqt lsure?e luaunire 1n;rarrod y rr 1rI3 lE.rrl, t0 rrrx

rlrot s turr.t I l0

'i(urlrucs asolc ptrets 1ou lul eapr slr{1 'aIqEuosaal aJE 1eq1 sBapr {ueu os 'Jaranop 'IIloloqc{sd ssuru aIII Jo aJnsBaul a[qBr[eJ B sB salr€ry taIJBru {cots ldacce o l e I ' l B u o s B eJ a ? r n b a u o l s u . r a a s ? I 'se.rn8rJ acr.rd {rols Jo sarres snoauatouroq aeooqo o? InJarBc sr auo Jr Illercadsa {cerncJB laar8 q?Tn sqt8ual alcAc ?uturrotap ol alqrssod sT tl 'acuarled aqr pue arult eql sBq auo Jr 'aro;araq1 'I11enuuu sB IT3a s8 ,(1rep pue 'A11aan 'AIr.lruou a[rlBIrBAs arB Iaqr ?Bql ur sarru3r; {ueu ralo a9elua^pa 1ea.r8 I a^Eq osIB sa.rn8!J ta{Juuf Ico?S

-s rpun {1aeIlslar arr o? ,..t]^^l?r.t::;:l

-a.l prn 'at!l Io por.rxl StpI (IrTeI B ra^,o pualx? uraq?Jo auns esnecel sarrSr3: lelrEru Irol s q?In S l u rryo ,rp a l l l sA u n l B a ^B rl I 'lJor srql lnolB ur sa I ot n o l l sa ra l u ! I I IA r ! Iu l rl t t '?aIJEur lcsts aq1 ur salcr(r eq? qr!,r tpl?c?uuoc ul errqBu {reururtl -a.rd B Jo l ro a Jo IB a p p o o S B o p 01 !,1 al'lTl aql PBq I vr6l Pu8 tt6[ ur )ov Ll

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Report for March l95l

CYCLES

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PrtLtr!riAty ?Gitooosrrr (0rrrs' DtLrA (Al vrtctrl Flc.f. ?or arou?t oF trocrs At frotcATro, JAtr.1932 - Mrv 1950. Prrrl trorclrg ?ossrrrE cycL€g At or rcAr rrE LGtctx rrf xoltxs Ag Inotclr€o. ilort ?t{A? THI cYcLCS tu6GE5TrDOtfrtr f x p o R T A t r T L yf a o x G i o u ? r o c t o u 2 . c r { A r r ! r € ? t c . t f t t r t o t ; , o f E. s, C. Coppocxor Srx &rroxro. lrrrs. ^ x o A R € R E p F o o u c r or x R o u c r r x r s c o u R t E s v .

llhar al I this addsup ro is that the traders in dif l'erent kinds of stock do not all l' ehav e th e s a rn w a y a t th e s a n e ti me. I r r ot h e r s o rd s , s to c k p ri c e s w o ul d seem not t o Le a n i n d e x o f n rrt p s y c h o l ogy. The most that one can say is llrat the prices of gr or r p s o l ' s to c k s m a y re p re s e n t the i ntellectual and emotional attitudes of the t r ader s in th o s e p a rti c u l a r F ro u p s. A gain , i t m i g h t re a -s o n a l ,l yl ,e s upposed that cyeles in the prices of stocks of e o m p a n i e so r f i r o u p s o f c o m p a n i e sc o r r e s pondedto the cycles in the profiLs of those conpanies. Fut people who have studied the suLjecc tell me that stock p r i c e s v a r y m u c hm o r e d i r e c t l y v i t h t h e sales thatr with the earnings of tlre cofli-

panies irrvolved. I therefore imagine ttrat w e h a v e l r e r e a n o t h e r e x a m p l eo f s o r n e t h i n g thaL " ougi l t " to Ie so, Lut i s not. From the alove olservaLions I assume tha0 che cycles w€ iirrd in the prices of s tocks rn di l'leren t companiesand di t feren t indusbries prolaLly corresponds to the c y c l e s o f t h e r r l r r o f c l r o s ec o m p a n i e so r i ndustri es, Ltt I do not know thi s. f r n a k e t h e s e c o r n m e n t sc h i e f l y t o p r o vide suBgestions for future lines of research that you rnay {ind fnritful. For present purposes w hac i s i mportant i s the fact that di fferent ki nds of srocks show di fferent cycl es, or the same cycl es with different amplitudes,or wirh diff e r e n t t i m i n g . E e c a u s eo I t h i s f a c t a n y

[n.Jare3 ,(1 epBrusAettlE esJnof Jo s r l uaut - l s n f p a s l q l ' ' 1 6 I S u r r n p a p e r ux e p u r a q l u r a i u e q c c r ?s B J p a q t r o J p a l s nf p B e r e . $ se.rnflr; sauof-,iql aqt Ir pernpa.r {lttari ar1 plnon ecualarttp stql f rr1 'x+uI sauof-noq aql puB xaPul salaq) eqr uasaleq 8! esua -ralJIp ?satearS arll .stxapur lrloul ?saq aq? Jo aeJql uaealal secu?ralttp aql noA s,noqs 'rllrpll lr cl S.llrrrtcl uoJt uorss tru -u'o? Belao3 aqt Jo uorssruuad q?!r pecnp - o.rda.rs r '?, '8!f arM sr;crq.trt .iTf 'xaPul sa[ nr)) aql asn of uole lcep ar{l pasnec ,{11rar lerl? lcBJ .ra?181srtll sB/tll[ 'x3Pu[ sal,ilo] aq? ur A1re1n'iar o.rou pur ll.rralc arqr readds salc{c aqt Io auoctsoeI tB,uoIlIppB uI .r '83xaPul uroux IIeil Io ulsr3tlrr],, ,a.roqr o1 paJJ3TaJ Pet?r?ua uo!?cas I uT r.IrPEI IrOls-uourD0lloal aql ur PaurslrDJ ars qJrqa Io euos 'suoseal Io JaquntrB rof xaPul uotssrrluloc salaoc eql esoqc I r r P r I s eI r l l - . 3 3 t ! l r 0 3 f q I 'Pr".f,lor tf$[ A.renuBfrlort s q l m u A q a l q u l r B A B . s 1 a r . r 1 s n p u l, A - d x e p -uI uorssrurDc salmc ?rl? uo sBn IJoa arll .Io r.red -ra1aa-riaqf ?ruf ,s[Brr?unpul Eauof aoo aql uo {ro^r euos prp I ra,|JB|lt{3o?s aq1 ur sa1c,(c a q l s s r P n l s { u r u I I o 'Irrulu4 1 -rrd pur llcltfclorr lltrru r! ;r gqlacql ,q plnoqr rrrrl Ilru Irrr rpErlrf qrlqr rfpEl PrllqEcr I uc {rcr lsv.aaorlB o? parJ.Ja.r {trauatouorl Io ),lcBI}o eielue^pe -s.tp arlt taslJo Al.reeu tou saop aSetuerpe

s T q 1 . r a A a , n o l l' p a z r r u r u r r u s n q l p u e t a s l J o Alrred aJE sarJtsnput Ienpl^rpur prrasarusd -'rloc [8npl^IpuI Jo suoIlErJBa IvruapI'Je e{l xspuT uB rl3fis ur ?eql l JBI aqt s r a3vlue,rpe sHL .slco.rs IIB Io xapur lr.ramil B q1I,l tttr1."O ur alBluBAPB Surlesuadrnc ,(l lerrrerl e sr ararfl ptrerl rayf o eq1 uO 'palnpot? ur i ura,1 A uet suor aJe saf ust slns 1r ^ r e u q r r r l , $o l u T p l n b r I J o 1 s a B J o s r s l l e u e 01 snoieIBrl Iof,tiuarlf, I aIBur o1 Surlrl sr sarras s qrns az{1eue oL .au!1 t{f .saJr{c IIB aJn?f,rd aqr olur iurunc 1u.raJ -I Ip a^Brl {eru qclqa .Io autos rslcols nau rFIa a1 plnoc xapuT qsraao e(lf ur salclc e r l l P a s n I u o f , / r r o r la e s d I r s u s u B f n o A 'U{bI uBql Jeqlrnt ou o8 sar.rlsnprr! aql Jo Jleq l1r1llno.; { t B , l ' xaPuI scr?s11 B?s PJePUPIS-uOrsSrtutuo3 sa[no3 eq1 oluI paJnpoJ?ut 3J3 slcola Io spuII snorre^ aql Jq sacrrd qrrrla roJ sel?p a$ snoqs qsrtlJil selBrcossv pu8 pJt selrq) parJIV Iq Irrrprl lr0lS-uqnf3 rlolf lJBrlJ e ( .3r4 se .iolaq acnpo-rdar I 'IlrauaSounq s T r { t r r o l n o q s o J . 'ratr8l alrlll B ro Jo {cet

J e r I J B e a [ ' t ' t I I e : l u r u p c s e l c A o ? u r E So r l l r o 'salcAc u a r a J t I Ip 11 pezrJalJererlr a1 Auru rlf,Irfi sleqs--3urrl o? e!T? u/oJt pappB uaaq aAsq sa lllsnPul f r r a uf o s r f J o ? s l s q l l J B f eq? 01 anp sr .(trauadounq Jo IoBI sIqL tlsturi

orto1 I ou rrltrtlv

I rUrH

:lt .l S

.snoareiotoq sB papru?e-r aq ?oulrc alorlri s sB la{JEur {Jols eql Jo ror^Bqsl aq? flurluasardar sa.rrfiry Io sar.las tuol oioirrxol

s8

IS6I qcrB6 roJ trod"U -

'l

'!lJ

S3-lf,AC

86

I I t a

CYCLLS-Feport

forlfarch l95I

rtt 3r(xt3 rrot6riAt3 triao^o' urn[t{9

a tsc(Lt^fa(o('5 J{rY|ctS t s|a?ric Ano 5,nPlnt Dt.tG a rr..E |rto ltrtttc.-|r5.t((r,|I.oi

3 Fl

E= rF G

I I a !t

3fctt lloaor. ft1.ror'r srsiIeuB alcric ur spuaJl aBe:a.le Suraour asn ,,{IIeJauat,, norCfBqf pTes 'papua uado sJaffeu o^{t tJaI a^eq I uoTlTsodxa a^oqe aql uI I

t III

aBe6 rrossarl

LeesouIII P*e I ASsl€ologn!. In adcl'lllon to atu(ytng rr,Vele Analysie: The ltovlng Averag6r d.ownto rC. lhe Usp of thc llovlng Averags trt Cycle .AnalXeis. tr read the folf oviag assJ.gunant ln Spun: cbapter

14:

chepter 15:

Pa€e zrj

through the ftrst

par€€raph on pege zgg.

Page 320 through the flrgt

paragrepb on pa€o 3zS.

Pa€e 327, ilOraphlc versus llatbeoatleal liethodg,n thqu€b seeond llne oD page 331+. $ote tbat tbe advantages and dlsacvantages of the novlng ayerago refcr to thelr uso ln cleterulning eecular trend, aot to thelr uee for cleternining eycl,ee. . Pa€e 351, rrsunnar.y,tt d.or.rnto nProblengn on paga 3Sj. Pfoblanrg l. Io further your stock rnarket study please novr co4ute a centcrcd 9-year lovlng avGre€s of the logs of the lncler. If you have access to a co4nrtlng nachlne uee the short-+ut nethr,'- :" 1 post your ansvrer ln Colrrnn C. Ii yonr-bve only a4 eddfng nachlne end. a slide rule post the novlag totale lnto Cohrnn C., port tlg of each of these totals lnto the next column ancl call lt Colum Cr (io te rcad C prlue). Eead Colunns C and Cr aecurately. f,ote thg,t ln eonlnrtlng e 9-year arltluetle novlng everage of the logs of the d,ata yor bavs ectually eouputed a 9-year geonetrlc noving av€ra€6 of tbe data tbogclves. Conelder the flrst teru of ymrr novlng avere€e: Hben yor aild.ed the uluc logg together lt vas aquivalent to nultlplylag all nine tuder values togetber. tfhen yotr illvided. the eun of these logs lt vas the egulvaleut of taklng tbe nlnth root of the produet of all theee numbers. 2. Nor plot on a sheet of }'t':r ; 17n arithetlc of tbe ttock narket ilata (S. & p. C. A. ,--Cbart.^j).

polylrurpole pepor the logr

3. llow plot, on the sane chart, uslng the Bamescaleg, the ceatered 9{eer uovlag erora€e of the loga of the d.ata. Plot the novlng average by neans of a 'broken llne. Do not s€nd this

chert in yet.

I r.rill ealL for lt

4,

Ew long dld. lt

take you to study thls

5.

Eon long atld tt

take you to trork Problen 1?

after

Ilesson fY.

legson and the aaelgnnent? Probleng 2 and. 3?

I

tg0t $ilIds '0t

I

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H>UVflSilU il1>A d@

flVNHM@K

9t' 'IJoA iraN'1u

1 J u , 1, r r . r t r

'su1cn3 rr Apn'ls .xlt JoJ uotl cpurru.lt.,r! 'AV | | o t t u ! t u . r t1 l t r t l . r l I U l I . r . r l J J l l ( r A f , t r . r q f [rll..l 1s'tli llrt)r 'rlJca s a l u o J a l n u t ( ' J a . r A e r { r ' Q r s t s o J r l . , J E r s q i i } I J A J J o I t l t l J i l ( f l ' . r l I t , | i l ( r l l ' ! | t . r > ,l l t \ 5i,'lr rJe

rol)anf0

nnlrrrlurf5 lrlrtrlr3 F. -u.!rs 'lltplttflfg rlollh ro?tu!{l llr-? lolnp ltarcS

'Ar111o? IITr.l 4I .sleacloJ '{aai'. 'ar3 'elo3:aroql .9uo1 a loqrl olld salcfc s{aen 9Z seql uralqed eq6 'uo plftt 08 rtE aql € p€aqg peqcuT efi.q TIT4 1T eo1c.(c $ auo? Btq 1T aulTt 31aer.r .se1c.f,c ortl uI solc.fc .f"ll oarql uT gj 1' pcaqo pa-{ouT o.duq TITr.l 1T ,a1c.rtc raci :i9an Z' trq peaqp seqcuT alcAo :leoA-Z'5 artl lcqt oaJa8qo TTT'|trlol .do1 aql Jal$a s4ealt uoltoq 9'Z eotroc '{aer'l puocag eq& lllolloq aqt snTBA a sYq alcjc Pr1ooos Jo 't)uoces eq,[ eql uT 9zo'z Jo do1 l€ anTa4 € Bctl Ptl'e SuoI e4aen 7'$ sT e1c^Cc

'5 Ptm tt uoTlTsocluaenlaq .rte6JTeq sTI€J (516't enT€^) elcltc aql Jo rlolloq TEnlc? aq[ leedar pur

596'r 996't 9oo'z iza'z 9oo'z

onTol alcfg

I n €, z r ,

rtaeil

ssTql o:ITI soot 1I 'Ilanurluoc

r se o??d A uosseT

t rydrqoJd

Leseon V Paae 36 Problen J. 0n xrlrlay, Septenber 10, L95l+, Eethleheo Steel closert at ?6 518, 0n ftonday' Scptenber l3th it closed. at 79. Inte4polete cloeing priees for September llth arrdlSeptember 12tb.

Problen 5.

Conbrne the follor*lrrg:

A trendl rYhieh etarts

et 2.000 ancl lncrsasea .OOJ per yoar, and

A 4 year eycle rith an anplttude of / OO1 atarting at cregt, cycle valuee ,v111 be 2.001 , 2.@0, L.ng, 2.0@, end repeat. PIot the result

on arithnetic

In otbcr uorcls

paper.

Hm Iong d.ld rt take you to study tbe leaeon? [ow long did. tt

take you to rork

the probloa?

Is the lesson uaterial clear? If aot, untlerllae geutences tbat rrere ftard to rrnderstand..

ln retl the phraees or

Do you r'lant more problens? Do you rant

barder problens?

Pleese keep us postecl ln regard to all theee natters urtb eech leeeon' 8o th&t ve can regulate the corrrge to suit your needg and d,eeires.

OEZ IO sl1 seu11 5 sauTl 9Z uT 4€dtter IITA ureffa4

'u1ete a I T J A a g u a r d ' . l e o 1 a q o u s r s T t { l J I i,r ul 9 u r 9Z uT tBsd.sI TITrd e{cA3

i'e o'5

z.s

S'51c7-c-ut.qoci[ ;38€J .Iaq?ou€ e{tlf

'leeda.r IIIr.t 'elortc Jer{?o '06T s1'airba saull 5 9Z uoT1auTqlroceql eloJaq g{eeA Jo raqurnu aql '9 ET rolcBJ raqlo ar{& '22 sT Erol@J aql Jo oug '8u1uu1taq oqt Jo q13ue1 aq1 -Io-lqTod aql IoJ 6{ae'1 eql ol Ptmol€ lat ol t, ualqo.rcl t:1 alcac {eart'-Z'9 'e1c,fc aq1 3:o q13ua1 -JTaBliYTBSg leedi.r ol TT;oi se4.rlt rT aulrl aql ?nq. gz ser1el ll aql lou sT rolca; eq1 elcdc T€uoTlce:;; eql JoJ +€.11ildacra.ai:I€e a.{t 6I sseco.rd' ar{{ .r{l3uel T€uoTlcBJJ Jo salcrtc 8ura1oau1 suorl}cuTqlroc aa€q nort ueqt't'i'ro11 's1eeda.r uoT,€ulquoc ar{l aroJaq sllluctr 8T a{si IITri rT .d1SnO;aqg -3uo1 sr{luo8 BT ptlB 'g 'Z saTc^c al'rTqlroc- assc Jaqlou€ aiPl 'ro pasn aq iTTrl e1c.dc qc€a saulTI

( 5r

slenbatT rl u . . rl , 0 f Jq. paPTATP 0TZ

( tz s l e n b a O T

( 0 eslenba

.IIOc:.d€{ TTT1{ sTlil l€rql ouo la(lufitu l{lucltr Ial.I'? aulTl +SJTJ aqq 3t sTt{,tr '3u1uu13aq Jo 1u1od aq1 1€ aq. TiT''l salc'lic aqX T:€ l{tuou '{+TIZ ott? lV

'O1iZsier.ba L eauTl Z sauT+5 sauTl €. r o ' ( Z s a u r l L ) s a u r T. (+Z o a u t x 5 ) : : " T t e ro 'tfT EauITl0T saul? e 'arreq en sru{6 :A uosserlJo'i1 a?eCuo .3u1uu13aqJo actld atil oi'u1eiil plo9rs ouo" 1uo111sod; o+ uoTtruTquocsgl Jo.,:pellnDe.r aulTl aqr eq. ITTr'i sq15ue1ot{l Jo aICrTlInI!souruoc ar€ salc{o ue{ri Jo spolred saao er€ salc.rtcaql puit paurquoc ls€al aq+ l.rt*

rrIrfis$

gau dleq no.riueg 'd1qsuo1x€Teraseqd au€s eql of punoJ€ euoc 01 saTcdce.IoutJo or,ll JoJ parlnbal aur+ ;o q1tua1 aq? Xno e:n313:ol ulaas l rnec I

snoTlssrs swlClc ei{IltrIaloc I lqeualtidng

n llossgl 1 a8e4 luaualcidng I uossal

L e E s o nv Supplenent I PeBe 2

fupplernenttnq thls

ansuer. on. etudent urlt es :

Pnobt atn: Deterntne the secondart eycleg thst of d,lfferent lengt'hs are conblned.

are generatecl uben tvo regular

eycles

Bul e (frol o3le.tronlce):

frequency arc conblnedl tvo nenr frequencles Vben tuo slpalg of cllfferent e^re generat et13 slgnals (a) a frequency cqr:nl to the sun of tbe two orlglnal signals. (t) tuo origlnal the of dlffersnee e freErency cqual to the

AEDII-

(Note that

cailoil-

e l$-nonth eycle bag a frequoney of Llt+ eycle per nonth)

A [-nonth cycle conblned wlth a 6-aontb cyclc rtll

gencrate-

a cycle havlnq the frequenc" of I l4 I Ll6 = 5lt2 vhlch neans 5 cyclce !n 12 nopths or a Z,4-nonth cycle' (U) ianlng the frequeney of lftr - t 16= LltZ "'"y"f" nealts a l2-nonth cYc1e. vhich (")

Srnrbrly a conblnatlon of 4 arld tponth eycles u111 generate a 2g-rnontlr cyele antt a 2.Z-lnoDth $ lZOl cycle. Conbinatlon of S ancl 6-nonth eycleg vtll and.e Z,7-nonth (rr/lo) cYcre.

prodluce a 3O-uonth cycle

rselclc lo 6iiTil[5' trorJ peluTrde.r qfucl

.glTriareq notr ?tras ?ureq s1 O96t raqtrocaq 9u1e:eaea o1 prnter uT feTralaE T€trollTppy

'e1a.fc luauoduoo reqlo eql Jo eqt no^( e^P TITA uollalnduoc 8Tt{il 'euo €ETil sepou ueeAlaq seaaA Jo reqsnE erp ooTet fq (6 defe Iq peuplqo ss) pol.red clpou.Ce eql ecTAl epT^Tg '9

'ee1ctrc lueuoduoc eql Jo eql rc.C eaTt 1TTA uollelndtcoc BTtl[ 'eso Fn=fd sapou uearr1sq sea€A Jo euo Jo qfuel raqurut aql eoTeq.6q (f, dale .6q peuplqo 8") polrad clpou.Cs eql eoTal epTaql 'ft 'oAl trq pol.red crpoc{a eql JIdTlTrril

.e

'polred o1pou.6s

eql 'e 'I 'epou ol epoq @rJ rr"ds eEIl eq1 notr salt IITA ernpco.rd sTrtg, q€uet I .Z (lr!e$dde) eepg eq1 trq euo dets .Cq pouTelqo sea€A Jo Joqtrnueq1 f,1dlllrqt

'e cnld q cn8 lreu eql ol (cpou eql) qTTEBAf,aql eraqA emld aelc{c

aql Jo qfuel

euo qorJ BeaEAeql lunoC

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Inspeetlon Countlnc lntenralg Thrurband.pencll Oradrrnted,geale T}e Ttrre Cbart , InrDae,tlnn & A licnnr g[ 9fJ.trnp lilntr OI Cy"J. &qtrelb. 1. 2. 3. 4. 5,

1.

Ths caslest vqv to ftnd. rttythrn ln the r4r srrd dotmg of a surve ir by rirgrle inapectlon' ![htr nethod taksg a llttle practlce but ln aany lnetaacel, I flntl lt adco-uato. If you look at Currs I tn Flr?. 2 (legeon V) lt shoulit be pcrfcetly clear to you by lnspectlon that the curvc fluctrrateg ln a rhythn about 8 nontbr long. Thc !at!s eycls rhoulil bc alnogt cqually apparent ln Curre G 1n the lano ftgure. llhe log Curvc (rt) of El:.-Jtr of thle lcaaon rqlroduccd Cu:re O of r'lt. z . (lerroo Y).- Selor lt (E) 1r the rans cunc vltb a pencll llnc to rhou uhat the tralncd eyc lces lmedlatsly. It ls roncttneo easler to loo!: for lovs onlx, aB tn Tlq. 1l+Curve C, or for bleha only ar ln F1:. lll Cspc D, As thlg eutJect le so trportant let ug loolr at lore cxarnoles. Trg. L5 sl'ovs Cunc 5 of rlt, 2 (LceeonV) as tt ls and,ae thc tratnid cya rocr It, elther fron looktnq fron lorr to lorr enrl fron hlqh to hlEh, Ecrcaftcr, to sayo lpace, I rhell ghor uhat thc cyc rees lorr to lov !t, hl8b to hlp,h. Iou rhnrld. ree-Itze borercr thst the cyclc analyrt ulll looh at cvcry eurrc froo loth polntr of vlcu a! ln F1:. Ilt anit llg. LS. Iou ean often cce ccncrel rhythna ln thc lanc cunc at thc lanc tlne. 31,?. 16 lg a tyrthetlc cutre nradeqp of s 9-year ctrile and a lpy-ear cycle ln conblnatt,on, fou ean lool: st elther eyelc at :rour ogrtlon, d.leregardlng the otbcr. ltbc ccparatton le very ealy trhcn one eycle lg ge,veral tiner e,s long as the oth€r.. l.trin the connonent rhythng arc of noic noarly the sane lcn3th, E! tn Cunc A of tr'lq.'L?, thc seoahtlon'lE-norc d,lfficu.It, tut ulth a llttlc practlcc lt tg not too hard. clther. Cnrroe D, C, anrt D tn Flq. l? ehov thc thrcc dlfferent rhytbe vhlch thc traluedl oye sees csatly ln thls sunc. Seelnq d.lffcrcnt cyclae ln thc rarno surre 1o aonethiug ll!:c foeuelng-your Gycs on dlffercnt obJeete ln the ramc vleu. Eold. your hand ln front of yorr nnd fosus oa lti the other sld.e of th€ roon le blurred.. Socus on the rrlnd,oucurteln; yout hand }ccornesblurred. Poeus on the vlev gtrt the wlndov; both tbc hand antt crrrtata becoac 'I3 blurrect . ts the rane uhcn yor look et one rhytbe aad.gut other rhythns out of your oycrr &r tt verc. Fll. L? ls lnteregtinq for another rensorl. Ig ls nrde q) gurely of randon rnrnhcrs. Tho obgenrod rhytlurs arc pGrfectly "q^1 tut they arc not !.ltl1jllcani. Thev v111 not contlmre. They ere horreror nonrlv ag re.nlar as the rtrythr ln Ourvc '; ln trit. 2, aira rnrch nore rerrlrr tlull the rhvthdr ln Csac C ln 81q. 2, wbere, la both tnataneao, ve lorov tbc rWtba ls elgriflcant, bocausc lt la thcre by conrtructloa,

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LessonVI Supplement Page I

Lissol! vI SupplementP IIAXII{GTIIIE CH.{TTSVOIU(HARDER Tbere are nar\Y\ays in vhich you can nake tine ch$t are }istail belotr.

vork hariler '

SoIDeof theD

l.Doublothobarshavln€doutleorlriPleyourstanclarilclcaraparr.Louger cycles are thus often sugge"ted. (See Lesson VI, p'rge 1l') use 2. :lhere you bave but a forr re1:eti'tions of a cycle it often helps to ag uell' bars not only for hii:hs anil for lovs but for uEraril anil ilovruraril crosgings IuseblueanilgreenforthBatlditional.bars.uherlyouilothlgxluhavgfour].inog line6 to average' across the titne chart instaaal of tlro. ThiE giTes you for:r slope Sonetines this helPs. cyclea ' 3. Tirne charts often enr-ble you to notice reveraing V-A.) of reversing cycles see Lesson

(!or a illscussion

length Plottcil on a €riil tt'ice or threa tin68 aa (See lesEoa YI' laea I3') oit"n illuminatingi """ chlrt of one len4th hi'ats 5. TinaDy cones the Batter of getting tron a time yet bsen fully covere'l' of cycles of otber leDriths. This subjettt has not perhaps

4. Cycles of a particul3r long as the particulcr t en..lth

su€gest6 the possibility .luf, tendency for bars to cranl uphi'l)' or dormhi'tI another cycle. Ior lnstance refer to ths tine chart lllustrate't of LessonVI.

by tho blue lrint'

of

pase 35

feds fron llote first of all the ilrastic drop off ( tovartl the rlght) of the the botton belov lines later c olunng ? the red. bar 3t nonth lf9 to ttte rett bai four of the qrid at Position 4r0. The pencil line falls off Connect these tr'ro bars l,ith a light pencll line' 2l lines in ri colunns, suggesti'ng a longth of 2) | 273 28 l/4 noaths' ilovcontinuethepencillineuprrariltotheleftuDlilitcutEtheto!ofthe griat (about a quarter of the vay int,o posltion 289)' l i o n t h n u t n b e r 2 8 g a t t h e t o p o f t h e ? l h c o l u n n i s t h e s a n e a s n o n t h n u l b eald r2S9 your llDe utrn'ard at thg botto'n of the 6th c olunn-. Therefore yo\r can continue (23 lines in itt" 6ttt colum, using the san€ sloPe to the left fron the t"ft"t-"i into the vay quertsr the top of the gri't about e 0 colunns) ' Your line,riff "ut position rePreseDting tlronth 1?&. It nill.run froo posltion Slnilr-rly you can extend your line to the lie'ht' poaition 5I9 519 at the bottoo' ancl a3ain fron 40& at the top dormvard t;-;;"i;io" at the top dornr'raril to -t,osition 6Jl' at the botton'

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Lesson V[ Suppl ernent I0 Page I

L'1SS0}!VI Supplement 10 A,'O{f}-r:R T Il rI CllART searce-'except

ec&use tiure cbarts aro rather

fevr cycle analysts trho knotr about thern--I think

of the

rn the frles

you vill

be rnterested

to

see a tine chart of the 9.18 nonth cycle in the ton uriles of the Canadia^n paciflc

Railyay,

Thrs trrne chart r.ras nad.eby G. lieredith

of that eompany. It

Statistician

Rountreer Chief

is reproduced from Cvc1es for June, 1951'

page 216. of actual ton miles,

The time clrart is of the deviations for

seasonal variations, Tou trill

note that,

from their

p-nonth moving avera€e'

to sirrplify

the presentation, tir. Rountree nade

a 9-uonth tiloe chart instead of a 9.2alloued the bars to faII

off

atljusted'

or 9.18-nonth tirne chart'

to the right

and fitted.

Ee then

a 9.l8-rnonth }lne

to then, Iou r,rill also note that, duetion,

b e c a u s e color r.ras not available

for

repro-

had' somo of the usual time chart conventions ln regard. to color

to be violated.. rt To fill Fou-ndation

uP the back of for

1951, reprinted.

the

this

page I aro reprodlucing for you the

StudY of Cycles lnd.ex of Rarlroad Stoek P r i c e s ,

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FOUNI'ATION FOR TTIESTUIIYOF CTCLES' IltltlEx oF ttAILlroAL sTOCt(pttIcEs, lgSl-l95l O construet an index of railroad commonstock prices srritable for cycle a n a l y s i s r e p r o c e e d e da s f o l l o r s : As the besis of this index we used the fol lo*ing aource raterial: (l ) For the period l83l-1F70 we used the Cleveland Trust C,cnpanylndex of Fail Stock Prices; (21 for laTl-1q38 re used rhe Standard and Poor's Corporation Index of Railroad Stak Prices as extended Lachrard by AIfred Corles IIJ and Associates and puLlished hy c h e ma s I n d e x P - 3 i n t h e i r l , o n k C o r r m o n Strcl ladrrrt; and (3) for the period l9fo-1a51 re used the Standard and Pmr's Corporation Railroad Stock Index as published in the Survey of Current ltusirrcss.

As the Standard and Poor's Corporation Index is nq hased on lq35-39=100, Lut vas f o r r n e r l y l . , a s e do n l a 2 6 ' 1 0 0 , a n d s s t h e Cleveland Trust C,anpanyIndex differs frcrn the Cwles Index, adjustmnts rere medeas f ol I o,r's: l$l-Ia?0, x I.lda l 8 7 l - l Q J R ,x 2 - e a 3 lei9-1q51, x 1.000 The values of the new index, thus derived, ere given l;elo,. You can keep this index up to dgce frqn year to year or month to nronth or even from reek to *eek frqn the Survey of Curr"nD llusiness.

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Teehnleal CYCLH ANALYSIS HOW TO CET IIINTS OF CYCLES Part I

In Two Parts T o m a k ea c y c l e a n a l y s i s o l a s e r i e s o f fi6lures, You must do four things: First you m,rstget a hint of a cycle that may l'e present. Second, You must find out aII you cRnabout this cycle. Third, You must @ n r et o a n o p i n i o n i n r e g a r d L o t l r e s i 3 nificance of the cycle and accept iL as e pro[ab]. reality, or reJect it as probably dre to randontl'orces. Fourtlr, You tnust see if tlere isa hint of another cycle presenL, ord if So, you ntust go llrrorrlh the same s t e p s a l l o v . ' t a 6 1 a i n .T h i s p r o c e s s n r u s t b e c o r t L i t , u e du n t i I a l I t h e i m p o r t a n t c y c I e s present have been discovered, studied, d r a r a c c e r i z e d , a n d e v a l u a L e d .Y o u h a v e n o w completedyour cycle analYsis. If, following your analysis, you adjust the original figures for the cqnl'ined efTecu of all tlre accepted cycles, the renrainder will be a cqnl,ination of the Erend and the sporadic or non-cyclic fluctuatiore ( includiitg in this category the unimportanL cycles, if any). To these figures you can fit a trend which vou can project cith due 'wrap' tlrediscovered caution. You can then and accepted cycles around this trend eo drow what will happen, except for noncyclic [luctuatiotts, if the trend and cycles continue as delermilred. You will recall that your first. srep is to get a hirrt of the cycles that may Le Fesent. llow do you do this? At che Institute held a! Foundation tleadquarters from June Sth to June l0tlr, E discussed fifteen of the ways in which this can te done. We also discussed types of nrachines that have been found useful in tion. minimizing conrputa la

An outline of this part of the workof dre Insgiuute follws: HOSITO GET HINTS OF CYCLES TTIAT MAY BE PRESENT IN A SERIES OF FIGI.IRES Gaphic l,lechods l. 2. 3. 1,

Simpleinspeccion Quick count Graduatedrule Timechart analysis

Computatiqr a I ltletlrods 5. Linited data 6. Autocorrelation 7. Relauives: (a) simple; (b) complex f . i4oving averages 9. Section moving average: (a) simple; (b) weighted 10. Eliminaticn of trend: (a) by ccnparison of the data with their moving averages; (L) hy other nEans 11. Comparison of one moving sverage with another .12. Periodic tables 13. llarmonicanalysis: (a) simple, and (b) multiple 14. Adjusc for cycles already found 15. Dr. NorLert Wiener's methods ltbchines l. 2. 3. 1.

Optical machines illechanical machines Electrical nr.ochines hrnch card and taLulating machines

It may prove of value to You if outline is expanded.

this

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LessonVI, Supplement13

The Time Chart (r:rtlter lf you arc pellornringcyclr analyslsrn:rnually lhan by conrputer)arrolheruseful deviceyou cen use lo Inspectdatr for evidenceof cyclesis lhe linre charl.f irtre periods(lenjtlrs) chaltrrnaylevcrlpossible ln addition.the lime chart will give an indicationof phase.but lt will not verify the velidity of tlre cyclc. nor reverl its precisermplitude or shape.lf r possiblecycle is irrdicated, rdditional rnrlysis must follow. Tht tirnechartwasirrvenled hy Mr. ('h:rpinlloskinsof New llarrrpthe Instituleof Trend Researclr of llopkirrtorr, slrire. The time cltrrl is r dirgrarrtof lrighsand lows sel up irr suchr wry aslo giveyou lrirrlsof rhythnrsthrt ntaybe prerent In your time series.lt is a most inrporlantndjunclfor cyelc anrlysis.lt dependsfor its uscfulrress upon an objectlvemethodfor delernrinirrg lhe "highs" and "lows" fol any given wrve hngtlr bcing souglrt.The lirne chart itself is then melely r nrctlrodof dirgrammingthesehighsand lows In such ! wey rs lo thlow light upon the lenglh and eonsisteneyof the rhythm or rhythmsthat nraybe preserrlin the figures. The efficaeyof the tinre charl leclrniqrre depcndsupon the eoncepl of clerrspan:il is necessaryto undelstand clerrsprnbeforcrttemptin3time charls. Flrst, the definitlon: clcarspanir r nrrnrbcrgivin3 the dislrncetnelsuredin time unils fiom r point In tinregoing beek to the prior point where.for risingvalues.llrereis a vrlue equol lo or h[her lhan lhe vrlue at the starling poini: for falllng values,clearspanis lhe distance(in tirne unils) brck to the point whele lhele occuti r valueas low osoT lower llran the valuert the startint point. Note that elearspanis a nuntberlhat counts rrnitsof tlme. lf r drtr point is rbove the immediatelyprlor dala polnl, clerrspanansw?r3 the question:llow long hasit been slnee the drta were lhat h[h? You count brck 3tl many lime unlts until you bump into a valueashigh asor higher thrn the one you stuted with. What is courrledis lhe time unltg,rnd the tnswet is lhat it hasbcenso ntenyweeks.or monthl. or whalever.sincethere occulledr valueas high (or h[her thrn] the one beingertmined.

Considerihe followingscriesof datapoints.

TimePeriod Number

Data (i,e.:value)

I 2 3 4 5 6 7 8

t02 90 98 9l 96 98 t0t 93

numberfor the valueat position6 is 2. The clearsnan Thevalueat posltion6 is 98, andthe valueis up fiom the unlil youreachthe previous one.Youcountbacktwoplaces value theclealsprn With position. rising e lhe third 98 at is a blackvalue,and is obtainedby countingbrck until you reacha valueos highas or higherthan lhe valueat r highervalue. means the starlin3point.A bhck clealspan position 7 is r black5, value al the for l0l The clearspan the l0l rt position lherebeingfive time peliodsbetween 7 rnd the 102at positionl. Whenthe valueof the datapoint is downfiom thepre' viousone, the clearspan is rcd or minus.A led or minus lor the meansr lowervalue.Thus,tlreclearspan clearspan vatueof 93 rt position8 ls led or mlnus3, therebcing threetime periodsbetweenthe 93 at position8 rnd the 4. 9l rt position is countedand notedbesidethe databeing Clearspan studicd;the time chart is then plotted from the clerrspannumbels. The timechartcanalsobe plotteddirectfylroma chart [itst counting o[ the figuresbeingstudied.The technique, numberspf r stliesof figures, and noting tlre clearspan then makinga time chart(or maklngthe timechartfiom tlre plottingof lhe tigures)canbc usedat anystagein the data, canbe rppliedto theorig,lnal The lechnique analysis. data,for example. or lo detrended the for recording The time chartis r 3rid constructed

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To checkif ihere is r 6-lermcycle.a 6-lermlirrrechnrt is constructcd.There ere six vertical positions.l'he first position in each column is zelo to that column and lhe posltionsgoingdown arc l, 2, 3, 4, 5. or 6 plecesafter. Going down the fourth column. the one headedby 18, l8 is the drte of the zeropositionor basefor lhat colurnn. The first positionbelow l8 marksthe l9tlr time slot; thc sixth positionbelowmarksthe 24th time slot.

llowever.crestsand troughsate more correctlyentered successivelyt the completechalt follows.Crestsareenteted as black lines, troughsas red lineswherecolor is possible. Hele a brokenline is uscdto denotelows.

F i g . 4 : A 6 . T E R MT I M E C H A R T O F A 6 . T E R MC Y C L E

F i s . 2 : A 6 . T E R MT I M E C H A R T G R I D The first high(crest)in the 6-ternrcycleoccursat time position4. On a 6-term time clrart, a higlr or low must havea clearspan of at least3 to be entered.The first high occurs rt time position4, and hasa clearspan of 3. lt is firllowed by r full standardclearspanlow, and can lherefore be enteredon the time clrart at position4, four placesafter 0 in the first colunrn.The secondhfh alsoqualifiesunder the rules-aclerrspan of at least3. end lowsof full standard clearspan beforeand after;it canbe enteredat position10, four placcsafler 6 in the sccondcolumn. The crestswould all be enteredasshownbelow.

The first entry on the cltart is a low at positionone. you cannot be sureit'is a low-no dataoccurling Because are placedaroundit on the beforethat point-parenthesis chartandan etrow noted. As mentionedabove,highsand lows are enteredsucces' sively.Althoughthe resulthereis two horizontallincs,the lines evenluatedonly as each separateentry was made; entriesjust happento connect. the scparate The horizontalline on the Gtetm tirne chart indicates the possiblepresenceol' a 6'term cycle; fulther analysis of the cycle,and to is necessaryto verify the significanc? lt is obvious, obtain a descriptionof its characterirtics. however, that you could reach a conclusionabout the timing (phase)of the cycle frorn the above time chart. Note lhat the clearspanis written to the right under situation eachentry as lhe entry is made.In a complicaled notation can be helpful. this The time chart can be plotted from elthel the clearspan numbers,or the chart of the data. When the chart is is checkedvisuallyesshownin the following used,clearspan enlargedsectionof our 6-termcycle. (See figure 5.) Our example is as regular a cycle as it is possibleto find. Unlessyou ate unusuallyperceptiveor are makinga | 2-monthtime chart of a monthly weatherseries,or department store sales,or agriculturalplices,it would of coursebc unusualto draw r time chart with the barsgoing horizontallywith suchprecision. ANOTHER EXAIUPLE

Fig. 3: HIGHS OF A 6.TERM C Y C L E E N T E R E D ON A 6.TERM TIME CHART

The following tabulationrecordsthe warmestmonth of the year at San Franciscofor the years l93l-1940. The month numbersare also noted, rs the time chart k lct uP

dols 065 I J.X3N.08s [tlo)rxrltla).oml)rNlrNlud0rs l j rrNr'((tooo'+( 08soroD 09s n'(A)sxl:(oe)gvrrNludoss (,t)tNl=n.t!l'loPs Itlq'=.\r31 ots 08soroc ocs

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)gvr rNludobt v3"lJ,,:(sc ..sNvdsu I rxlN 08t lll>lrla.tsrore l+f=f l:l'l 0')[

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'selcAo . 3t{I si.lluoE 1I'g PaFcTpuT t uT tqluot8 ,2.6 "{PuoI o{l uT t{.. aql 'tL'Z uolllsod ll uollTso(i TtI lE Puocat Jlrxf 5t. BT o:'cddTls er{I '. rdi o1 ocualaJeE JTsq lsl aql uT q..T.{ eqt suolllsori Po.{loutsTiil 1t.{1 3Ao.[30t ZI a>cal X uo83e1

Leseon X Pane 13 2, llo rlles as to tho best r.rey of nnnlprlrtlon e:rn.be given beealtge the rclntlonghtp betr'reen the eyclne ancl ranCone ctlffer fron gerles to aerles and wen fron It lg, therefore, clear thst the natter of eycle to eycle vitl,ln a einqle gorles. posltlotllno cycle ls cul nrt and. not a $.lgDc,n , Iou ore not;ret far enou:h alonl1n'the courae to enable you. to exerclse this Pa.rtlculnrly, you needl to naster at loast a forthconlng art to the best aclvanrtage. ':rerlod-lc tabl.e bolerson on nwln.i: arrera,qesandl arrother fortheonln I lesnon'on the fore you er:r have'the prlnclplea fulLy ln rnlntt. Fotre;ver, I have trletl to nake then as elean as I can, bceed on vhat you nov knotl. Cgn!:'lrrrloryl I have trled to enlarqe your horlzons reqard.lnq ths problenn of posltlonlng thc I have tlone thle bv dwotlnl a legson to the gubJcct of nlnlnizing randons, cycle. It ls not as alrlole as lt nay seetn' Posltloninl be done accurately. a cyclo eannot Therefot6-l,

Alrrayg keep your fln*cre

crossed ae to the aecuracy of your cycle poeltlon-

lng. Z. In gcneral, as gtstect ln I'cgson lX, uoe utrtnardland d.or'mttardcrOseln4e to nlonBtt of lat antt Znd hnlveo of your perlocllc tablc. Iocate'rhl,3hen anrl ). ord.ir'to

In qeneral, do not bother to overlap sectlons of your periodle beiluec rcndons . .

tabl.c-'ln

b. In genera! do n6t uge fancy nwin: etr€raies, guch as novinf, netliarre and ucl...hted norln€ Everages, to anooth the wera'?€B of the halvee of yor.rr perlodlc tebleg. . periitttc table by a 5, In scneral, do snooth averaqcs of eneh aeetlon of a novlng a?eraqe about a thlrd or e hnlf of the leaeth of the cycle, dspeniling upon ln the wernq€8 tend to cone l/J ar tlz eyclc lntenarg' rhether the lrrenrlrrlties

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L:lSSoiix Prohl enc

Problar 1. lry a 5-north novlnq aver*..e of the 8vorfl.?cF of the va,rio.ts cohl!|ns of .ach h&u of controrl'cd 'ta'tlr Irooted into llabre 3ll' l'!1"1' 1t the lgn:t'i' t'f evele lnillcateil by thls Rethoat Problco 2. ?osltlon each balf of cach of the perloillc tabler courtutril ar lrart of Lessoag VII o l VIII usin;i ths nsthoils tlleeosseil ln Lcscon X. Iletemtne tlrGor-oarevith regulte bbtaincil frot! tiDo clj,'rt analTJts ittcatett cycle lcnqth, anil you prorlous jrosl tlorrltt ?. Probleo 3. llelre lev periotllc half of eech aev tab1e.

tables of the lnillcateal }.n?tha enfr rsporrtlotr Ra-iletenrlne each cycl e lenqth.

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