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FUND AMENTA NOVA
T H E O R I A E
FUNCTIONUM ELLIPTICARUM
A D C T O H E
Ka'N Csu=>Ta.<j j4V<.o\» Xac-o'^*
D. CAROLO GUSTAVO lACOBO TaCOBI,
PROF. ORD. IN UNIT. REOIOM.
REGIOMONTI
SOMTIBXIS VRATROH BORNTR^OER
1839.
PARfSlIS APCD PoKTHiiv & Co. TnscTTit & VMbks.
LONniNI Apuo Tmvttil, WtrKM & Ricrtib. H. W. Kottmi. B»n, Yora«« & Yomc.
AMSTELODAMI aptd Mtretum & O. C. G. Sdiipk.
PETROPOLI AP«n G.xi... ^g,,^^^ ^^ GoOglC
C-'-
db, Google
P R O CE M I U M.
Ante bienninm fere, cum theoriam ivmctionum ellipticanun ac-
curatius exammare placuit, incidi in quaestiones quasdam gravis-
simas, quae et theoriae illi novam faciem creare, et universam
artein analyticam insigniter promovere Tidebantur. Quibns ad
exitum ielicem et propter difBcultatem rei vix expectatum per-
ductis, prima earum momenta breriter et sine demonstratio-
ne, mox cum vehementius ilia desiderari, et invento novo vix
tidem tribui videretur, addita demonstratione, cum Geometris
communicavi. Urgebar simul, ut systema completum quaestio-
num a me snsceptarum in publicum ederem. Cui desiderio ut ex
parte saltem satisfaciorem, fundamenta, qnibus quaestiones meae
superstructae sunt, in publicum edere constitui. Quae fundamenta
nova theoriae iiinctionum ellipticjaiim iam indulgentiae Geometra-
rum commendamus.
a* Digitized by doOgle
Vt a typographorum inendis, quantum fieri potuit, mundus
evaderet liber, CI. Scherk curare voluit, cui ea de re yalde me
obstrictum esse profiteer. Quae emendanda restant, ad calcem
adiecta sunt.
Scribcbaoi m. Fehr. a. 1829
ad UniT. Regiotn.
db,Goo';5le
INDEX R E R U M.
De Transformatione Functionum Ellipticarum. §$. 1— *4. p«g. i— 85
ExpoiJtio probUinatw. geper^lU de traqdapnatioiije. }}. 1. 2. ...,..,.. p«g. 1
Priucipia traufaimftintU, f^ fl. 4. , .... . . ■ — S
Proponitur ezpreuib -^— ^ — '— — — -'
e tmufomurtibiie «xpr«ub)tuB
/±&— )tr-(!)(
—■>)(?-!)
Forinam simpQcioretn ledigenda
iL
- In sKaua eita simiiciD -
55. 10 — 12
Proponitur tranaformatio tertii ordinu. §$. 13. 14
PropODitiu trenaJurmatio qointi ordiois. $• 16
Qaomodo tranrfonnatione bis adliibita perTenitur- ad mul(ipUcatiou«m. $.16.
Dfl nolatjone nova fiuictionum elliplicariun. $-17
Connnlae in anftlTai functionuin ellipticamm fundameutalea. $.18
De imagiiiariia f'uaotionum ellipticamm raloribut. rrlnoipiuin dnpliois periodi. $. 19>
Theoria analytioa transfonnalionis liuictionuin ellipticaruio. $. 20
Deinonstralio Ibrinulanim aualjfticarum pro trenst'ormalioue. $$. 21-^23
De vnriii eiasdcoi ordinis tritnsl'onnatioiiibiM. Transfonnatioues dune realei , inaioris moduli
in miuorem «t miuoris in maiorem. $. 24- v
De traosfonnatioDibus complemeiitariis s. qnoinodo e transfonnatione moduli in modulum alia
deriTatur coinpleinenli iu complemeutum. $. 25
De traDsfoTioationibiu suppltmeiitariia ad multiplicationpm. $$. Z6> 27.
t^ormulae aualyticse generalrs pro multiplicatione funttioiuim fllipticarum. $. 28>
De a«}aatiouum rnQdularium ailVclibus. $$. 29 — S4 — 66
Digitized b, Google
— 28
— :6
— S9
— 80
— 88
— S4
49
— 5T
— «0
— 64
TheoriaEvolutioDisFuDCtioDumiilllipticaram. $§. S5— 66. pag. 84 — 188
De erolulioue I'unctioaum ellipticanim in prodacta InfUiIta. (. SS — 38 ptg. 84
Erolntio futictionuin eUipticBruin in seiiei ncondum rinaa vel cosiinu multiplorum tiga-
meuti progredientes. ^. 89 — 42' . . , — 99
Formulae generalei pro functionibui sin am | -. 1 , — ■ in leriea evolTen-
^ ' ' •'»"'"(^)
dia, wcnndum abau Tel oosIdiu multiplomm Ipains x progi«£enteg. j$. 48— 46- ^ 115
Integralimn ellipticorum nenuida species In seriei evoMtur. $$.47. 48 -— 1S3
Integralia eUiptica tertiae speraei indeflnita ad catum reTOcantur definitum, in quo ampli-
tudo parametiiun auquat. $$. 49. 60 — 137
lotegralia elliptica tertiae speciei in seriem erotrnntut. Qnomodo ilia per tranicaideDleBi
novam 6 domtno^ exprimuntw. {$. S|. 02 — 142
De additions argumentorum et amplitudiniA et paiametri in tertia speeie iutegralium ellipti-
oorum. $$.68 — fiS — 151
Keductionea expreuiuniOn Z(tu}, @(iu) ad argumentmn reale.- Beduotio geimdis tertiae
.apeciei integrallum ellipticorum, in q^oibu* argnmenta M amplitndinis et paiametri
inaginaria sunt. $$. 66— '60 — ]gt
Functionea cjlipticae anat ftmctionea fractae. De fuaotiooibiii H , 6 , quae nomeratoria
et denomiiiatoria locum tenant. $.61 .'.. * — 172
De erolutione fuiMtiouum H, G iu seiiea. Evolutio tertia fuitctionum eUiplioarum. De-
mousttatio onalylica theoreuiatis Fprmatiani , nnumquemqiie numerum esse summam
ijiiatuor qtwdratorum. $j. 62 — 66 - — 176
,y Google
TRANSFORMATIONE FUNCTIONUM ELLIPtlCARUM.
EXPOSITIO fKiOB(.enATI$ OS«KRAI<IS PE TpANS.rpH«*T10NE.
Intesntia maxime memorabilia, quae Formula ^-itliilu'iitnr / ■ e rT»..4.aul ti'&vmc^'Vuotf
tioDum Ellipticarum , quae dicantur, primam' ^jMJtenk- «oiuritaaul, ab Argnmeato du-
plice pendent, et ab Amplitndine $ et a Modulo k. Eiusmodi fuacrtiiri^.inlirjae.ia^ipar
ratis raloribns, qnos ilia pro divertis ' Aw^ilitu^n^aa- <^itHi> ,+ eodem maoente Modulo,
egregia multa defexerant Analystae/quaA AddiTi5neit)'%ortii& t^rMultiplicatiouem spectant.
Quam naper riflimud i^aaestionem a CI: j4t'il'iaCi>mmBatini6M; tmifttti l8tfd^iuii^o^,''ini-
Tom in moduni prorectflni 'eSse (T. Crfet/fe JdufniaY 'Mr .viAaif tiftd' aflge^ab'Ae'MatiieMti^
■ il :
Alia est qnaestio nee minoris momeliii-'^^iB)m0 -^evsu tatissimo capta illam ioTol-
renft — de comparatione -FviWIliMiMn:. BUiptic^miii ; pro> .fifadtdkri. iiHbfaH»^> dmr^
Quern qnaestionem post praeclaia inr'Ail&iCf Xf^« Af'Tj —^ !7ii^riae Fuoctioniim EUipti-
camm Conditoris — ad principta ceDa Dos'primi reTOCavi'mus, eiasqoe solntionem dedi-
mu5 generalem (V. Aatrononiitche tfachricKun A. 1827. No. ISS. IJE?"). ''' H^tic nostfam de
Transfbrmatiooe Theoriam et quae alia uide in Analysio Functionum Eltipticaikim' redun-
dant, iam fnsias exponemus.
oy Google
2.
Froblema, quod nobis propouimus, generale hoc est:
„Qua4ritur Functio rationalis y elemettti x ejutmadi, ut ait:
Qnod Froblema et Mnltiplicaliooem Tidemtu aipplecti et TraDsfbnnatioDem.
iDDiiinera iam din coostabent exempla eiusmodi fnnctionuin rationalium j, quae
problemati proposito satisfacinut. Frimum notum erat, quicunque datus sit iiumerus in-
teger impar n , einsmodi fanctiouem i-atiooalem y exhiberi posse , at sit :
/A+By+Cj'+Dy»+liy' /A+B.+Cx'+Dx'+Er* *
<|aod est' de Moltiplicatione theorema/ Oueni ib finem hdhtberi detiet fdrma :
t + >'i+«V + »'"it' + h ■(nn)iM
Coefficientibua a, a', a", . . . . ; b^ b% b", , . . rite determioatis. Satis diu etiam ex-
jdfu^itapL w\t .fonuun lumc:. ,
ae^'hBxuf'gnttvaikonA-. ■'- .1 • •< ■--' -■■ ■'■>:■. ,■'.,■.■
:: ■'■.'.., ' :: •■■ >«4,.'»(«fV-M:M>».> .■■^>>"'»»'° ■ ..
. ..,,... ,,,-^~ b-W>>rtTl|V+»^V + -:' + ''"'°y°/ ■: ;; . *, ',
qwe ex iiUps subsiti^tionis; repetitiooe oitum da^it, ^ta determinari posse, ut solvat pro-
blema. . Ngy^ admodnm etjam proVaftum est a CI** ^^^dre , eum in fiqem adfail leri posse
Ibrmam banc rite detenninatam:
MB ToHoBy taAmm substitiittoae repctita, haoc gtmniumm:
b+b'i+bV + k"'.' + .. .+b(»"'<'"' *
His inter se iiuwtis loriiiis patet, problemati aatisfieri posse, idonea Cuts Coefficienliam
eleotione, posito:
^ .+.'.+.V+.".'^-...+.<".P
'* b+b'r+bV+b'".'+ . . . + b*«i' *
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liquideai p tit n— fgn> fonuu 2«a''(S.m-i-l)*. lam r«qiieiilibw prabaLitnr, idamT*-
l(a«, ^ioiMfue «itf p rtwiMnM.
PRINCIPIA TRANSFORAIATIONIS.
3.
DesignenlDr per U, V fnnctiones radooales inlegrae elementi x ; sit porro y '= — ; lit :
Sj VJU— UdV
/'A'+B'y+Cy'+U'j'+E'y* /y '
Lreritatu caaaa pouto:
V At] 1 7 il V
Fractionem -^ in fbrmam aimpliciiweiii redigere licet, qnoties Y factores doplices
hahet i qnin sdeo , nbi praeter quatuor fectores lioearea inter se dirnsos « reliqnonun nu-
mero faini inter se aeqnales existant, firactio ilia sponte in Differoitiale Fnnctioius Kllipti-
cae redit ■ ■ ' ■ ■ , desienrinte 11 faoctionem elemeoti x rationalem.
11/ A+Bi+Ci*-|-Di>+Es*
Qa«n accnntfns exemii^emiu oaanm ac rideamns, quot et qaales sibi poscat Coaditiones.
Sint fimctioneaU, Vah»ap", alteram" ordinis, itaatni<p: eiitT(4p)*' or-
dinu. lam nt, qoataor factoribns linearibiu exceptis,' e reliqnis fimctionis T fectoribua,
qnornm est nomenu 4 p — 4 , blni inter se aeqnales eradant, (2 p — S) CoDditioniliiu
satisfadendom erit. 'Qaot enim fiinctio proposita dnpUces habere debet Jactores lineares,
tot inter Coefficientes eins intercedere debent Aeqnationes Conditionalea.
At fanctiooibas U, Y Qoantilates Constantes ladetraminatae ipsnnt m + p + 2»
sen potius m+p+l, <|nippe e qnarum nnmero nnam aliqiiam=i ponere licet. Qua-
rum igitnr nnmero rel aeqnatur numenu Cotiditionom 2 p — 3 vel ab eo superatar , modo
snpponatnr, m esse aliqaem'e nmnerisp — S, p' — 2, p-^t, p, qnibils casibos nntatfrns
ladeterminatemm fit resp. 2p — 2, 2p^l, Cp> 9p+l. ' -IhuM priores casus reiicien-
dos esse cua infra demonstrabitnr, turn hnnc in modam patet. ^arnqne inrentis fimdio-
nibns U, V , qoae futctioni Y formam iUam jwacflcriptaip oonciliatit, nhi loco x sulwtitui-
tur«-|-0x, neqne ordo mntatnr fiinctionum U, V, T, neqne nnmems foctorum dupli-
cinm faiictimi& T: node, ia solotiDnem inTmtnw statini jditaa Qnantitateft Arbitrarias in-
Digitized by Google
feite lk«t. ' iM^tie t6iMa<B8 lilddteMikiiiatiirAiri ubiqeraifi Coniitiaaam <iwl>A* stluni uin-
tatibos superare debet , nndrwsus m=p — S, m=p — i reitAmdi t^Mii. Pwfr vid«cniS)
loco X poaito ' - ' , tertium casam ad qiurtaiai redaci et qaartum mioime matari, quo
igitnr caau Indetermiwaiarum ti^s et drbitraride man£tft 6t iaLbttm iriiimt
lam igitar erictom est, qwmtnin qi|tdem e numero iDdetenniaatamin et uuuiero
CondiHonam inter se comparatis concludere licet, quicunque ait p nvmerut, fortnam.!
j„ ■+"+'V-H-
.+-'
{■9\f
ita tUtoTninari pone, ut ait:
ztr
detignaate M fimctiofttm rationaUm ipuu* * f into tolutionem trta QuantittUts jirbi-
traridi iMttdvtri poaat.
Ul detennioetiir fimctio Ola M, sit T=±(Ah-Bx-4-Gk*-^Dx'-|-£x*>TX, de-
sigDADte T ionctioneBi elaneati x in^nm rationainn: erit .
.. . . T
Ipsa T eritoTdiiii9(2p — 2)**; nee maiom eise potest V-t——U-^.. lam casibus qui-
bosdam constat, scilicet obi niimOTUs p formam illam habet 2«S'^ (2n+l)% M adeo
fieri Constantem. Idem generaliter probetbttur sequentibus, quicnnque sit p numerus.
Fnnctiones U, V snpponere possnnuis &ctorem communem non tiaberej .adiecio
enim liMton qoiaqHwi, fra^tia y=^T'°*''* ii^uUitiif.. Resoly^mw expressionem
A'+B'y+Cy-fD'jr«4-Eyt
in ftctor^ Uniiares, itd at sit:
A>B'y-fCry+I»'J'+Ey«A'.(l-«'y>.(l-j»y)(l-Vy)(|-|*y),
unde iBtiaiU: ■ ■
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lam existere Don polMl Actor, i|u quantitatilHH \—mV, V'-jSV, V— >U,
V — h'V vel omniliiu vel imo daibns tautam ex eamm nnnwio conunimia til ; idem eaim
et V et U simal metiretiir, quaa fiictornn commnaem nen habfre nppotaiiacis. Itaqae ubi
factor aliquia linearis fnnctioiiem T bis metitnr, idem unam diqium e qnaiditatilbu V— alJ,
V— /STU, V — y'V, V — i'U et ipsam bis metiatnr Decesse est.
ian qoteptar aequatJonw seifiiMite*:
,v_.-u/"_iiIi:iH),u.v 1H_«"
,v_^,iH_iiI=£fl.„_,.iH_oiI
,v-i-u,"-"^-»">.D,V.lg-p".:
dk 4i di dx
e quibuaseqnitiir, foctorem qui nnam aliqattm e qnautitAtibiu V— a'U, V— jffD, V— yU,
V — SIT bis ideoune etiam eicu dififereBtiale mftiatiir ^ eaudem metiii expressioaem
bus, conflatnm posiumDs=T, mide T ipam V-r U— metietur. At T inferioris or-
dinis non est qaam ipsa V-^- — U— , nnde videmns
abire in CoDstantem.
Ceterudi adaotemiUj ubi lunctionum V, V altera ioferif^ts ot^inis feiMc* qoam
(p — i)", ipaam etiam V^ U— inferioris ordiais fnisse qnamT, quae tamen Jku
metiridelMl; ^vodcwnabsardui) ait, reiici debduat casus mssp — 2, m=p — 9-
lam igitor demonstratnm est, formamt
Digitized by Google
ijmcumpM »H numtrut p, ita ^ttUrminan pottt, tU prod§at:
dy di
V A'+B-y+C'y'+D'y' +t'y' /a+B.+&'+D.'+E.»
QtMKf «•< Prinoipium in mtmria Traruformationum Fuactiooum EUipticarum Faada-
mtntale.
PROPONITDR EXFRESSIO ^^ — IN FORMAH
V ± (y-«) (y-fl) (y-7) (y-5)
SIMPtlCIOREH REDIGBNDA
5.
Trium Coiutantiiun Arbitraiianim ope, qnas solutionem . Problematu nostri ad-
mittere nditnas, expressio A+Bxh-Cx'h-Dx'-hEx* in simpliciorem redigi potest haoc;
A(i— x*)(l^k*x*}. Ut hoo et reliqna, quae mododemoiislratasaDt, exemplis etiam
monstreator, propotitnm sit, datam expressionein :
/±(y-«)(y-«(y-7)Cr-!>
facta snbstitatioDe:
in simpUcionni transfonnare banc:
Qnaeiitar de substitatione adhibenda, de Modulo k et de factore Constante M e datu
quautitatibus »j 0, y, i determinandis.
Fonatur a+a'x-|-a''x*=U, b+b'x-|-b"x'=V, j=— : e principiis modo ex-
po&itis fiuri debet:
(U-«V)(U-flV)iU-TV)(U-iV)=K(l-.'}(l-k*.^(l+m.r(I+n.r.
deaigoante K CoDstantem aliquam arbitrariam. Hioc ndemas duos e namero faclornm
U — >V, U — |3V, U — yV, V — SV, qui enrnt secondi ordinis, adeo fieri quadrata.
PoDamus igitur :
Digitized by Google
lam c^aod reliqaos attioet &ctores U — «V, U — |8V, pouj polerk, aut:
U — «V=bA(1_.»). O— flV»B(l— k»«»), .irt:
U— aV = A.{t-ft}(l— kx), U-|9V=B{1.4.x)(l+k>),
designaQtilras A , B, C^ D qiuBtiUtes ConstSBtM. Friiu niiciendwDi erit. Frodiret etdm
■ |^ - ^|' ^ = — ~* =— . j^?, i , uode seqneretor, ekmento x in— x mutato y inunutatam
maoere, (|uoi1 absurdum esse patet ex aeqaationibiu :
u-
^ T— T
A
I-*-
u-
7v^
=7^'
A
(l+n.)'
Pom igitur debet:
« U-.V=A(l-.).(i-k.)
t) n-(!V=.B(l+.).<l+k-)
4) D— >V=D(l-|-i>4'.
Adnotare coaTeDit, e Goiutantibiu A, B, C, D uoam aliqaam ex arhitrio detenninari
6-
Videmnaexaeqiiatioiie 1)} etpoaitoxssl etfiouto xs=-^ fieri U=«V. Hinc ex
aetinatione :
u-ov B'(l+.)(l+l.)'
poaifo x = l, pndit:
posito x^— ;
.-, c (■+f)'
.-« B
tmde;
db, Google
Frorsns simili msAo inTenitur:
mde n=Vli, Bs= •k. Keqao eiiim MipulM po^ere U«l m et • ; hipi euim cxpressio
il^|l=i-i^, Meoqnt ipM y •fciret in CcMtMl«n.
lam in aeqnatioDe:
v—,y T-; c I j+A^ I'
ponatnr primnin x = -*-l, quo caso C = .V; deinde x= — I, quo casa U=8V:
prodennt duae aeqnatione* sfquenles:
a— t £ 1 1— v^ i ■
Ouibns in « 3ucli» aequalionibus, fit: ' ' '
O V (.-5)I(1-J) '
undo pouere licet:
Ce/c.-tXU— n • ' • I
D=/(.-5j(0— J) i
nam e quantitatibus A, B, C, D una ex arbitrio detenuinati {i^tsrat- .
Ex iisdem aeqnatiooibns, altera per alteram divisa, obtinemus:
l+>^k _' /(«-Tl(g-')
V"(«— r)W-^-y%--»)"M-irl
'/(—r) (»-') + /(— ';'«-">
Adnotetur adjiuc formula :
, /(«-T)W-Ji + /i"— ')(a-Ti
^ /(•-«(»-') — /c-'iis-
,, Google
unde;
Ut Coostaittes A, B, defiuiantor, observe, ex ae<(ualionilms l), 2), S), positox=-^,
qno fiicto U=8V, erui: _
,-. *<--v^'('-/4)
'"' ♦/(•-7).»-S)
^ -V(.-T)l— ') j/(._.w«->i-v^(— Ma— ,1 j
E urincipiis generalikus uipra a nohis stabililis seqnitur, in exemplo nostro expres-
sionem V-^— U-i^- aeqMlem fore producM (1 -t- /kx) (I — /Ix) in qnantiutem con-
stantem dncto, quod ita facto calculo comprobatur.
Fit, uti evolatione fiicta constat:
!(acti antem sumns:
V— JV*D(t— y^-")*.
unde;
J(»— fVi^ tc(i+y\..)>^
J(»-»V) ^_tD(i_y\..)v-t.
Digitized b,Goo';5le
Uude prodit :
(,-!)(vi|H.-U-i^\ = 4-4y-k.CD(l+yk..j(l-v^l..). . ( '
His omnibus rite coUectis, obtinemus:
£y -t-iVy f CU d» ■
/-(y—)tt-«<j— >)(?-') '-' ' -"*' /ct-«'l(i-k''") '
nnde :
-j-J A=AB /(.-TlOi-»r-/(»-8)W-rl
"=T4vrV "CD— j;;^^: ^
/-(J— JlJ-WO— »)(!->) m/(1— "Jd-k'O
Posito (.—,).(«— S)=G, (.— 8)(i3— 7)=G', iit:
M. /(!--■) (■-!■■»•)
\rr-T>{ ^+'-^ i /•/G--y^y_.
Sit G=min, G'=Qn, sit porro:
in'= — (m + n), ii' = /mi
m"^— (m'+D*). n'^V m'l
erit, posito x=Sin(P:
d9
Oternm valor ipsius x facillime computatur ope formulae:
l+Vk.i V t«-*)W^^'V y— r ■
ubi: ^^" = 77: T?" V — ::7^:r' —
, Google
8.
Qoaatitates a, 0, y, i in foimolu propositis ex aitiitrio inter se penualare licet.
Qaod itt arbitrio nostro positnm certum fit ac defiDitnm, simnlac conditio addatnr, ut,
siqnidem fieri possit, tnuisfonnatio per buIntitDtioaem realem succedat. Id qaod acca-
nitiiu examinemus.
Fooamos, qoantilates «, $, y, h reales esse omnes; sit porroa>3>T>S) ita ut
a—fff » — y, » — 8 sint quantitstes positivae. lam distingaendnm erit pro limitibna, in-
ter qaos Talor argumenti y coDtinetnr :
l)itty. t) v el 0, S) |S et «, 4) « e( t.
Cagu postremo traDsitum ab « ad t per infioitum fieri puta. ExjUvsuOQeoi - . '■ ■ ■ ■■■
non nisi casn secundo et quarto, exprestionem _ ■ ; . -■ ■■ ■ non nisi casn
/ -(y-«){y-WCy_T)(y-8)
prime et tertio realem fieri videbias. Sabstitntiones reales, quae quataor illis casibus re-
spondent. Tabula I. indicabit. Deinde Tabula D. formulas amplectamur, quae expres-
sioni — ■ ^ - per substitutionem realem in simpliciorem transformandae in-
V ±Cj-«)(r-/J}(y-r)
serriunt, pro limitibus, inter qnos valor argumenti y continetur :
Quas formulas diridendo per i ac turn poneodo 8^— do facile e Tabula I. derirare licet.
Digitized by Google
TABULA I.
/(J— )(y-«(j— iJCy-l /"l— ■ /l>_N'.'
, y (—■ »)W-S"+ '/ ('-ffifa-»)
L= .J ^
.. /('— >)(H-i)-V(— m(T-»)
2
„„ /(— T)w-<) -V^(— »)ai-T)
db, Google
TABULA U
»' (r— )<y-s)(7-T) /i-." /!>_»•.'
ni^- '/"«-/!
y— T
II. Limites > . . . . P: , _.. = .,-
_dy_
/-(y-.)<y-m(y— >) /l-.' /t^
I. Limites /3
L_K. y"(.-,)(fl— ri
l+N. V «-» V --y
db, Google
u
9.
In tbrmnlu hisce pro limitibos assignatis simnl x a — 1 luqae ad + 1 atque y ab
altero limite ad alteram transit. Limitibns antem , qui formalis I et II respondent , inter
se commntatis, expressioni ~ Tid«Alu» v»lOTiew imaginaruim creari formae i»R, po-
.sitoi=:i/ — 1, ac designante R quantitatem aliqiiam reatem ; ipsi x autem conciliart for-
-=-itang|-.
Formam, ad quam hac occasione delati snmus, x= . in expressioDe a - .- i.(.\, i{
substituamns. Prodit :
/■<.-.■><.-, ^./(.-4^)(.-.-^*) /(.--")(■—'')
d^
/l-tka>,t<p+lk /■{! - k)' C05 Ip' + {1 + kj* sin V
Quae nobis qnidem substitutio satis memorabilu es.se videtur. E qua etiam generalior for-
mnla fluit sequeos , ponendo x=siD«|>:
/l — k'«n4'' /"l_fkco»a?>H-kk ■
uode pro limitibns o et n- obtinetur, evanescente parte iniaginaria:
w « «
,/ /"l—k'.ipiC' y /"l— «kcoi*ip+kk ^ /"l—Skcoj^i+kk
quae est demonstratio succincta formttla« meniorabilis & CI. Legendre proditae. B Ta-
bulis I et II daas alias derivare licet seqaentes, commntatis Umitibas, inter quos valor
ipsius y continetor , ac posito x = -TfTT' ^"^ limitibus assignfttis angnlus (p iftde a
e usque ad w crescit, dani y ab altero limite ad alteram transit.
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15
T A B. U !L A III
"Jy ip
Ay— »){y-./3>(y_v>{)' — J) ■ V^mnieoi(|lV°n«<>(p'
m = V'l-—rm-Sf('>-mr-t>
/"(»-T)(|i^^'^-/(..-ffi(.,-»)
V — Cy — ")(y — jSJfy — v)0' — I) /~mmeo(^*^nn,in^r
■""VC"— »)((i->>(--»)t«— t)
. .y(.-»)M-i)-hA«-i)»i-T) •
II. Limites
♦. V c- r)(— y /EJ
, Google
t6
T A B C L A.. IV.
ir '±.
/ty— «){y— 3)(T— t) /"muicMp'+a
I. Limites T . . . fl: H-j-^y ;;;r:7V 5^
' y"("-(i)(—' >)
a.
... „ . » /('-tXH— r)
I. Limiles — 06 ...•»: Wx" j
U. LimUes |9 . . . .: i«-r"V p=V V i^'
db, Google
Fuflius hBDc qaaestionem tnctayiimis, at ftdsit exetdplmn elalNiratam. Restant
adhuc casns, quibas (jnantitatniu «, B, ft S v-el duae rel qnatnor imagiaariae sant. Ca-
sus prior «t ipee sokttioncm rwdem admittit, quae tamen specie imaginarii laborat. Ca-
sus posterior eiasmodi solutionem realem omniDO non admittit. Qaare at otnoia ad rea-
lia revocentiir, novis transfbrmatiomLns opas erit, node cODcinnitas formularam pent.
Coi igitnr quaestiotii gnpersedemus.
Salistitationi |)ropositae alia respotidet, eias inT-ersa, fonnae
.= ■+■:'-'-•'/.
quae et ipsa formaks elegantuunuu sappedltat. Cum vero fortasse iam Qimis din'haic
quaestioni immorah Tideamar, eins iurestigatiooem ad aliam occasionem ralc^mus. Re-
vertimur ad {jHaestioues geaenileft.
de transformatiome expressionis
' aliah eivs sihileh
10.
Vidimas, datam expressionem:
M/l— x» /I— k'x
per substitutionem adhibitam hniasmodi:
^ b+b'.+bV+....+b<P>Mp V •
qaicnaqae sit namems p, id aliam eiiu similem transformari posse:
Eiasmodi substitatio cnm a datis Cotiffioientilws A', ,B\ C", D\ E' pendet, lum
Tero tnaximt a nniDflro p, qnippe qni exponcntem dengaat iligaitatis auiniiiae, quae io
foDctioaibns ratienalibiis V, V ioTemttir. QaamoliKm in aeqoeDtibns dicemns, eios-
Digilized by CjOOQ IC
modi snbstitntionem s. Imkafonoationem />'' ordinia ettt «. ad ff*"* ordiMm aive »impli~
cifia ad numerum p pertinere.
lam indolem hanim sohstitutionum accnratina examinatari, musam feeianuu for-
mam illam conqilexiorem:
ac qaaeramtu de simpliciori faac . , ad quom illam revocari posse et vi-
dimtis et DOtam est , iu aliam eins stmilem — __ ,. '_, ■ traiufomianda.
QaaestioDLS propositae Datura rite perpense, problemati satisfieri ioTeDitar, siqni-
dein IttnctioDmD U, V altera impar, altera par esse statnatnr, id qaod iam exemple in-
nnoDt ab Analystis hacteaos explorata. Qna in re maxime distiD^endom erit inter ca-
sum, quo iroparis functionis ordo paris ordine miuor et enm quo meior est paris ordine;
slTe inter casnm quo transformatio ad nomenim parem et enm quo ad numertim im-
parem pertinet.
Iam igitur primum proI>enm&, transftametionem snocedere adKiljita substitatioiie
■ordinis paris sen formae:
y- -(>+.V+."-*-l- . ■ ■ -H..^"-" .'■-') U
Hie (imctiones Th-U> V — U, V+l^U, V — XU et ipsae oimt ordinis pans,
onde pwiamas:
I) V + U=.(l+.)(l+k.)AA
X) V — a = (l— .)(! — ki)BB
S) VH-XUeCC
4) V— XU=DDi
designantibns A , B , G , D fnnctiones elementi x rationales integras. Quibus aeqnationi-
)>us simnlac satisfactom erit, emetnr, uti probavimus:
dy i» «
Mutatoxin — xcnmUin — U abeat, V aatem non mutetnr, ex aeqoationibus i), 8)
i«Iiqnae 8), 4) sponte floont Ut aequationibits l), 8) satisfiat, V-+- XUm ricibos,
Digitized by
Google
V-t-U (m — l) ricibiu dooi inter te aeqnaleft kabere debet fiKtores tioeares; iQsuperipsi
V-t-U etiam fiictor i+x auignah debet. Quae omnia Aeqnationes ConditioDales aibi
poscnnt Bnmero m-f-tn — 1-t-l^ am, cj^tti et ipse est nnmenui iDdetermiDatamm
a, a\ ... a^"'''*}^ b', L', ... bt"). Unde proUema propositam est delenDinatam.
80cuado loco pn^wniiis, succedere etiam traiulbrinationem , adhibita uibstita-
tiooe hniasmodi:
_ ,(.+.V+.".«+...+«"'U"°) TJ_
cjuae ad Dumenim imparem j>ertmet. Hie V+U, V— U^ Y-^XU, V— A.U et ipsae
aimt imparis ordiois, unde poaamna:
I) V + U«=a+')AA
t) V — U = (l— i}BB
4) V— XU=(l-ki)DD.
Hie quoqne solanunodo aeqnationibas t), 8) satisfacieodam erit, quippe e quibos mu-
taodo X in — x dnae reliqnae sponte manaDt. Ut illui»atis£at, etY+U, et Y-f-XU sin-
gnlae m vicibus daos.inter se aequoles haWant factores lineares necesse est, quern in finem
2 m Aeqnationibas Conditioaalibiu satisfacif^odnm erit, qnibos una accedit, nt insuper
Y-(-U nanciscatur (l +x) factorem. Hinc niunenini Aequationnm Gonditionaliuin esse ri~
demos Sm+l, qui etipseestnumerusladeterminatanun a, a', a", ..at"); h\ h", ...bC*");
node et hoc casn detenuinatam eat problema. .
11.
Designentur per U', Y' funcliones elemeuti y integrae rationales eiiumodi, ut
IT
poaito 2=-^, eraatnr:
di dy
Sit ea, quae adhibita est, substitntio 2==-^ ordlnis p'"; ac per aliam sobstitotionem
Digitized by Google
K»
yzsi—f (desigoantilni* U^ V, utaupra, faoctioiies elemsBtix ratioaalu iotegran,) qaag sit
ordiuu p", eruatur, nt supra:
J?
FA
lam soLstituto ralore yss-sp- in expresuone zs=-^, luucatur z=-rit : erit una ilia suhatitu-
tio z=^, (|ua adhibitft ernltar:
ordinis (pp')*^. Ita ridemus, e pluribns tramformatioiiibns , quae resp. ad nnmeros p , p',
p", ... pertiDeDt, successive adhibitis, unam compoDi posse, (juae ad nunieriim pp'p* - • •
perdneL Nee Don vice versS, qnod tamen in praesentiarum non probabimus, traosforma-
tionem, quae ad nnmenim aliqaem compositam pp'p' ■ • pertinet, semper ex aliis suc-
cessive adhibitls componere licet, quae resp. ad nameros p» p', p"> ■ pertinent. Quamobrem
eas tantnmmodo investigari oportet transformationes, quae ad numeram pertineant primum,
qnippe e qnibus cunctas componere licet reliquas. lam igitur to sequeotibus missum fa-
ciamus casom primum, qui ordinem transfbrmationis parem spectat, qnippe qnem sem-
per componere licet e transformatioDe imparis ordinis et transformatioue, quae ad nume-'
rum 2 pertinet, identidem, ubi opus erit, repetita. Casum secundum autem seu transfor-
mationes imparis ordinis iam propius examinemus.
Videmus eo casu fuocliones duas, alteram V parem Jm** ordinis, alteram Uimpa-
rem (2m-i-l)*' ordinis ita determinandas esse, ntsit:
Iain dico, ai quidtm ita functionet U, y determinentur , ut loco « potitp t— abeat
U 1 V
7=-?^ (/I -— = 7^: aequatione* ilia* alteram ex altera tponte aequi.
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PoMraiu V;=i^x*)« Vjeax^in.'); ridentu expreieiouein yss-^^s^- Joco x po-
sjto -r— abire itt ■ ■ - ■
ubi x^'^Fj-rT^i, ^^'^I'f'Ctri] ^9** fanctioavs iDtegrae. QmoA at ae<jaale iiat expressioni
— = T-Yj^-^-^^pr-, seqaentes obtiaere debeat seqaationes :
desigoante p qnaDtitatem Constantom. 'Ubi in his aequatuNiibiu mams ponimus t—
loco X muicucdinur:
Qoihiu cnm [sionbus compantis •M^a^oqil^as, ol>tiii«m|is ~j^=-rt QQ^e
Hinc fit : .
qaarum aequationum altera ex altera sequilur.
1 qnobes expreimio:
fjaadratiim est fnnctioius elementi x intc^rae rationalis, idem etiam vale)»t de alia, quae
Digitized by Google
22
ex ilia demattir praendo — looo x ac multipUcando per x*"*/^Xk*i"~i . Qoo fiicto ob-
tioemiis, tiqttidem — -■ - — quadratum ait^ Junctionem!
../T?=r!(^lH!Mi
= i+ki
= J+k. l+l. •
et (pMm guadrotom for«. Q. D. E.
Itsque eo rCTOcatmn est problema, ut expNuio:
Quadratum reddatUT, desigoante ^(x*) expressioDem huinsniodi:
Fit antem, posito U = x F (x*) = x (a -»- a' x' -I- a" x* -f- .. . -»-aMx*»>, cam sit
U=xF(x')=-/'^^x"'+' (p(-jiT):
I__ /-- b'" ^, AT b'— '' ^. AT b'"-'
.w_yz. k- , .<— 'i=yT.bi»-, ,"— •=y^I.b-k— . . . .
lam ad exempla delalumur.
,, Google
FROrOKITOR TRANSFORUATIO TERTII ORDINIS.
13.
Sitm=l, qui est casus sunpUcusimiu, V^l-^b'x% U=x(a+a'x*). Fosito
A=;(l+sx), enumils:
v+u=(1+.;aa=i+(1+i.).+.(«+.; ,•+.".■.
Hioc 6t:
b'=.(t+.). >-l4-f «, .'=.■.
Aequationes X $■ ^2 "> sequentes abeuat:
node obdnemas:
Poii«tnry>=u, y>.=r, erit.=i, l-t-8«;= •+*"' , .(2+.)=.^ilH^±^. Hinc
aeqoatio ;
'+•""'•+"1/5
abit in aequeutem:
1)
«•■
-»'+t.»(i_i>'o=o. , ,
Fit pnetem:
b'-.w.,=..(ii+i)='"-<'+-''
(.+«.■).•+.■•■
Digitized byGoO'^jle
M
Praeterea obtiuemus, quia. (-1-7;=. — —^ :
4) l-y =
(H-XT+ti'l)"
"+'■"■(»+«»■)«■
d-.X,-,")-
A_y A— I V— n'
6) /l-jj.
,.+.'u'(r+«a'
Porro loco x ponendo cum y abeat in —^, eruiraas secjnetitiijm for-
mularum systema; ._
, . ^ (H-.)(l+u„)-
„ , t'— ■)" — "!' ' ■
10) /l-
l+.u'Cv+t.")^
14.
Posito V-)-U = (H-i)AA, V-<-iU = (l-tt>CC, V-»»j=(t — x)BB,
V — )iU=(l — kx)DD, Tidimps fieri:
ABCD=M)v4i!._uilj, -. -i. .. + •,-•. ■
designaote M quaotitatem CoDstaotem ; qoam ex niuus^eiusdetn dignitatis Coefficienitis ooni-
paratioue, in ntraqae expressione ABCD, Y— 'U~j — tnstitnta, ernere licet. lam
posito V=b-|-b'x*-t- etc., U^ax-|-a'x'-|- etc., in siogdUs expressionibus A, B, C, D,
fit (^onstaos /V> nude in prodacto ex iis conflatb ^bj -ih txprekiam atitem V -3 U -r-
CoDstantem fieri Tidemus ab; nnde:
d by Google
Bine in exemplo iiostro,fit> quia bsxsl, a=-
»+«■' — »(«»-t-') .
unde;
MofloU k, X, t|uos per aequationem qnarti gradus a t>e invicem peodere ridimiu §.13. l),
foeik per eandem qoantitatem « rationaliler exprimantur. E fonnulis cnim supra
allatis;
-(«+•.] ■04 •) ■'
»= — i l+t.
Kqnitnr:
, -'(t+«) _..
Filinsnper: M j-: . , nnde, posito yssinT, x=sinT, aequalio:
V^l— y»/l— x*y' M/"i— «' V'l-k**'
in sequenlcm abit:
/(!+««.»-.(«+.)■ ".!•■ /l+t.-i^B+.Jml'
/■(l+J.)' C^-P+d —)■(!+•) •i-'T' ir(l+!.)Co.r+{l+.)'(l— J-taT
ad quam perrenitar subslilntioiie facta;
sh, r = t'+«')"'T-f...i.T-
db, Google
PBOrOinTUH TKANSrORMATW otimVi oHbims. '
■■ ;■- ,15. -~7 •'
I.im ad exemplam, (jaod simplicitate proxinium est, transeaniDa , in quo in^2,
Eniimuii:
uMe :
Hinc nauciscimiir:
Aequationts }£ §• IS fiunt:
■=/?4. -/!-■ ••=/?■ ■ •:
Ex bis sequitur: i '.:.'-
»'»' _ b'b'
sive, cam habeatur b'=(2«H-/3)H-(j3,-4-«»i), a'=|8(l -+-««)-|-(|3H-ai«):
t«+fi ~ flCl+«») ' ......
uode :
Hinc facile setjnitur:
(juod evolutum ac per » diTuum abit in :
Hanc aeqaatioDem his etiam duobua modis repraecMtare licet:
(""+«("-««=«(«-«)(■+««)
(, .+S) («_.)=(— Id) (I.+fl),
db, Google
nnde set|oitur:
His praeparatiA, reliqna facile Imuignnlur. Inrenimus eptra-, pai^tpk^a*, X=v*:
in + fi V _ h'b' __x^ ' il ' '
unde etiam:
«— 1/3 "" u* ■
Eat insuper (8 = »/a"^Y T"^^ — ' "^^"^ aeqnalioucs:
«• l« — tfl/ 0(l+t«) • « — J;3 u' .-,-;. ,
in seqaentes abenat:
sire: ■ ■• ■■■■I '■■'■• ■ - I- ■■■'^■■■■- ■ '■" I ■■-"-| ■ --- — ' •:. -m.
8oit{i— ov»)=ii(t«— n«) ; "'I '■;■■
«(TV + «i.)=«a'(l+ii'v), . .^ J ,j ^^ .^ ^^ ^.
uade:
Facta cTolutioii'; proflil: .■-.-. / - ■■
1) ii«_v«+5u'v'K— *'>+4uKa--V»'j=o. ...'.'■; .''..,
Reliqaa its inTemantnr. Ex fle((uatioiiil>us :
J«»Cl-l.T') = tiCv«-U«) , ,_,
seqnitur:
Hinc fit:
•—=4(1^)
, Google
■ .■,+8= '-'f';+'" ^u-(-iirj;-).
Hinc tandem dedncitur:
y=p+...H..+^= °"'t:?I'.r°''
lam cum sit M= — =7( — _"^ ], transfbrmatio qainti ordiois coQtiiMbitiir theore-
mate seqneate:
T H E O R E M A.
Fosilo ;
T(Y-U-). + u- (■■ + »■)(»-■.■).■+.'• (l-ll.') J
1) ,=-
, Google
pUOMODO TRANSFORMATIONE BIS ADHI81TA PERVENITUR
AD UULTIFLICATIOnEH.
1&
lospicieatem a^natioiies inter n et ▼ , daohns exemplis propositi^ inreutas :
fugeie noD potest, immntatu eas manere, abi v loco V, loco u autem ■ — t pouilar. Hiac
e theoreaute exemplo piimo ioTeoto, Tidellcet posito:
n*— v«+*B»(l — u't')=0
fieri:
idtenun statim derirator hoc, posito:
■ (■■-tv')y+v«y'
B«+«.Tt(„_,^j^ •
fieri:
/r=?.
lam vero est:
ande aequtur:
Utloco —8 wtiatlir + 8» sirez in — z, sive x in — x mulari i1el>et.
Simili modo e theoronate^ exemplo secundo proposilo, alteram deiliutlur , TicFe-
licet poaito:
'•M+b'tj-i- »*»(«•+»■)£"+»•) r*+»'''(«+'')y'
d by Google
lam cam setjuatar ex ae(|aatione;
<u.p,»)(v-u') uTfl— u'T*)— (g'-T^
{l + u'v,(l-Mv') ,..(l+ii'*)(l-u.') *'
fieri TiiieiDiis:
Ira traDsformatione bis adhilnta perretiitar ad MultipHcaliouem. '
Haec duo exempla, v't z. trausfonoationcs terHist qirinti'ordJDis, iam ))riu» in lit-
teris exhibui, quas mense lanio a. 1827 ad CI. Schumacher dedt. • V. Nova Antron. I. I.
Nee non ibidem methodi, qua emta saot) g^Kratitatem pTaedtcabam. Alteram biennio
ante iam a CI. Legeadre inTentum erat. ' < ■ ■ *
DE NOTATIONE NOVA FUNCTTONUil ELLlPTlCARUM.
17.
Mi&sis factis quaeslionibus algebraici^ accuratius inqniramus In oaturafn'atial^ti-
caro functioDom noslramm. Antea auteni notationis modnm, cuius in sequetRibaR mus
erit, indicemus uecesse est.
If)
Posito / - — ■=:u. angulum i^ ampUtudinem fnnctiouut u Tocart Geome-
•Terunt Hunc igitur angt
r:
(p = am
P' J.
/ — ,_ - ^Uf
trae cODsneTerunt Hunc igitur angulum in sequtolibns tlenotabirndsper: ' ampl. u seu
brerins per :
(p = am.u.
Ita, ubi t r - "/' " ~ ^"r *"* =
x=;sin.am.u.
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,81
losuper posito:'
vocaJjimoA K — u Complementain functiouls u ; Complementi ampliludiaem deaignabuaiis
per cotim, .ita nt Ht:
ExpreMtooem V^i-.k*nn*MRH= — ■ *^" , duce CI. Legeodre, deootabimus per
Aamu^y I— k'riii*amD .
ComplemeQtnm, (juod Tocatur a Q, La^udie, HodHU k deitigHibo per k', ita at sit:
kk+k'k's=l.
FoiTo e ootatione nostra erit:
Modulus, (jni snblDtelUgi debet, ubi opiu erit, aire uocis incltuus addetui , uve ia mar-
gine adiicietar. Modulo qod addito, in ' seqaentibus eaudem obi^ue Modulnm k sul>-
intelligas.
Ipsas expressiones sin am a , sin oomm a, eos am . u, cos co&m u, A am a, A coam u,
c«l. ac generaliter fuactionea trigonom4trietu •a mptit u di a i t , ia aetfuantibas Functionum EL~
UpticaruM nomine insignire couTCnit; ita fOf ei aomini aliam (juandam trlbnamus nolio-
nem atque hacteuus fectum est ab Anal^stiii. Ipsam n dicemus Argumentum Functionu
EUipticae , ita dI posito x ^ sin am n , ns; iifg . «n «m x . fi nototione proposita erit :
■tot an u . .
mh coam-*-^ — . - - -
k'riauna
dbyGoo^^jle
St
formulae in analtsi ftlnctionum elupticarum
fukdahektaxek
la
Ponamas am.n = aj ain.v = b, am(a+T)=ff, aiii.(u — t)s=^, nol
formulae pro additione et subtracdone FunctioDom ElliptMaramJandanentales;
■ coi b A b -t- fin b coskAa
l-k'«n
..'.inb'
co*ac
Mb — do
..IHDbA.Ab
1— k»uii
■'■■■lb*
a»&b
— k*iini
slnbcOfiCMb
l-k»«a
.•dob^
«iiiac<
MbAb-
• <iDbcaiaa>
I-k".b.'«»b'
eo«>M.b-|>«D
ia»inbA«Ab
1— k'.
. . _ A«Ab+k'«iifAilieo«ic«>»t
1 — k-iiom'iuib'
Ut ia promtu sint omnia, {juorum in postemm nsus erit, adnotemus adhnc formu-
las scqaenteSf qnae facile demonstrantur, et qnanun facile augetar nQmcrus;
t) Hn « + (in 9 s ; p . ,
«, - . - a. *a«.Ab
8) A <r + a » —
4] iiD <r — lin » =
5) cof ■ 9 — CM » =
6} a } — & «^ =
7) mh r . sin 9 =
8} l+k*ii07.tiii»K
9) l + dii,.d«» =
1— k'lina'Miib'
»iinbco»aaa
1 — k°<ina'uiib*
Xainaunb Aadb
1 — k'daa'.aiab*
Sk* iia a .mib coia ■ CM b
1— k'daa'ainb'
1 — k'aiDa'dnb*
ah'«^k*Nna*.cMb*
1— k'.ii>a*iS*b'
CMb'+aina'ab'
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t+KOtg.COti= -
U) i+a,.A»= i-k'riT."
12) t~k'ui)r*iDa=
a.'+k'jiBb*co«>'
l-k<>m.*iinl»*
15} 1 — 111) ff iin 9 = --
. 1— lt',in.',iiib*
... , ^^ - iin«'Ak* + iiiib>a.'
I*) 1— usrci».7^ — — p-7 — - ■ -
15) t — A <r ^ » :
ma*co»b'+«iiib'c<
1 — k'»ina'MBb'
«6) (l±in,)(l±A.9, (^.b±.i„.A g_
I— k «na°»mb
18) (i±k,i„.)(i±k»,?j^ ^tltw^i;-" -
19) (I±kiiiiffXlTkriDj;= .
») Cl±cM^)(l±«..a)c .
«) (l±co.ff)ClTc<MS) = -
{A«±kwtibco»a)'
1— k'uBaSinb'
(co*a±co»li)'
(.in>AbT"''t>^ii)'
««) a±<io(i±Aa) = — . ^'l':'"^''^',.
1 — k'uDa iinb
M) (l±i.)(lTi!l) = '■'■■'.'•.=^.'''
- 1 — r itn a >iD b
M)
b + »in b cw b, A
^ . ^ lia a CO* a A b — tin b cm b A .
96) lia ff & 9 = -
S7) HD » A <r = ■
28) coi •■ i 5 = -
S9> CO* » & (r = J
1 — k*tm>*ii«b*
coi b sin a & a -^ coi ■ tin b A b
COfbiina&a — coiaaiob Ab
I — k'wna'rinb'
oaacoib AaAb~*k'k'»iia(inb
T-fc'^a'wJTv -
Mae<MbAaAb + k'k'»maiinb
£
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M
SO) ™(,+9) = - ■ ""^
SO >in C<r — ») =
S!) „,(,:(.»)=-
SS) ».(,_S)=.
1 — k'«
DF. IMAGINARIIS FUNCTIONUM ELLIPTICARUM VALORIBUS.
rHINCIPIUM DUFLICIS PERIODI.
19. .
Fonamus sin (^ =z i tg ij' , abi i loco ^ — 1 positnoi ««t more pleris(f ue Geometric
1— k'
■na'ibvb' .
2*iati
.coibAa-
l-k'«n.'-.b'
eo.a'-
.«>a>*Ah?
l — k'
iua'anb*
CMb»-
-jmk'Ai'
\A-if
Hiocft:
id^i \A-if
Quam e notatione nostra in haoc abire videmas aequatioueai :
I) Mil .in i B = i tMlg ■« (u, k*).
HiQC setjiritiir:
f) cotam (iu, k) = ucait>(a, k")
S) UngamCiu, k) = i>iD.m(u. kY
4) a .m C 1. k) = ^""tu.fc^ ^ __ '
Co«aim(u, k) fiaci>aBi(ii, k)
i)J
uDCoamCiu
'■>=^»<„,i',
6)
coicoam(iu
,k)=.i-^eM<»..(u.k-)
?)
Ig co»m (i u
u -'
' k'.in™(u. I')
8)
A coatn fi u
k) = k',i„.o,«(u. k').
Aliud, quod bine flait, formnlantiti systema hoc est-:
9) tin am 2 i K' = O
10] *b *m i K' = cc , tc) li pbcci ir i X .
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11)
.U,«n(.+SlK')"=+.i«tm.
«)
wtvMi(a+ii1C)'=A — cofamu
t)
aaiii(ii-t-t>K') = — daaiu
»)
/ . f .7^ 1
15)
iAamu
— ik'
"""'° + '''''= k.l<,.,„u -
kcMcoamu
16)
1 ■ v +'
ig .n. (« + . K) ^ ^^ ^
m
d .m (u + i K') = - i colg .» II
18)
.in con. f.+iK') = .'^'"° = -
1
19) c(»coani(u-|-iR'} a
+ l''i
50) iKCOun (u+iK') = -^Aainu
51) AcMm(<t.(-iK') = 4-ik*tgafnu.
£ forniulis praecedentibus, quae et ipsae tamijnam *'r-"1"imrnitalrn ia Aoidysi fiiii-
clionum elljpticaruin cousiderari debent, elacet:
a. fuRctioDes ellipticas argnmCDti imagiuarii iv, Moduli k transforraari no»!(«
in alias argamenti realis t, Modali k' = /"i_kk ; und'e ^eneraliter fuoctio-
nes ellipticas argameoti itnagioarii u + i*', Moduli k, cotnpooere licet e
functionibns cllipticis argumenli u, Moduli k et aliis -argamenti y, Mo-
duli k';
b. fiiDctiones ellipticas duplici gaudere periodo, altera reali, altera imagina-
ria , siquidem Modulus k est realis. Utraque At imaginaria , ubi Moduluit
et ipse -est' imaginariirr. Quiod Printipium TtdpHcia iPenWi'niiDCnpabinius.
Equo, camnoirersam, quae fiogi potest, Amplectatur Periodicitatem Ana-
Ijticam, eincet, fuDClionea ellipticas non aliis adnumerari del>ere traoscen-
. dtntilnu, qoM^oibusdaiii gandent eleganlii*, 'f<H4a»»e i^rtbd* iHa<i aut ma-
ioribas , sed speciem quandam iis iness^ |?efecti et absolnti.
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THEORIA ANALYTIC* TRANSFOEMAHOiyS FUNCTIONUM
ELLIPTICARUM. i +
20.
Vidimus in aatecedentibus, cjaoties functioaes elemdini"x. ratioDale^ iutegras A, B,
0, D, V, V ita delennitientur, utsil:
V + U = Cl+i)AA
V — U = (l— »)IJB
V+^U=(l+kii)CC . ,
V— XUs=(l — kiijDi),
tJj ,.
poRito y=-r- lore: ■ ■(■
designaote M quantitatem Coustantem. lam expr«i»iioiie.s illaruQi fuucliouuni analjiicas
geofraleB pfoponaaias.
Sit n nomerus impar (juilil>et , slut m , m' nuiiieri iot^ri ^uilibet po^iitivi xeu ue-
gativi, qai tamen factorem commaDem noii hal>eaiit, (jui el ipse aumerum n metitur:
poiianiDs . , .
_ mK+in'iK' . ' .
'"~MV Mil'™*- / V rin^>m8<. /■■'r iii.'.in«(n — 1)- /
V = |i— k'MB'iift4«."J a— k'»Ui'amqM.»»V- T • (i — V.>ia'»mHlat~.t)m.tA
A = fiH — : — ^-r-) (i+-^ — ^—s—) ■■■• (^+— !, ., )
*" y tincauni— } \ lincoainSM / \ fiilcoanit(a — I)* /
C = ^l+kMacoun4N.ij /l+kt<DCi»m8(*.>l . . . ^lH-kjiacMii>«(o — l)»,<j
D = |1— kMBCo«ni4a.il {i — kMiicMm8«i.«t .... Il — k Hncoiint(ii— l)a-i|
,, Google
^ ^ h limcoain4a.*iDCOain8a . ... (uicaamt(a — l)"]*
~^' ' Inn ■m 4-iIb am 8«....riD «m » (n — 1) •/ '
OtiitiiisposiUs, ubix=:5inaniai, fit j=~;=suiam[-j^ , XJ.
Oui
Antequam ipsam aggrediamur formularum demoostrationem , earam iraDsformatio-
nem (jaaadam tDdicaMmiis. Qoem iii'fitiem liequealea aduotauius formulas, quae stalini
e formulis $. 18. d«cafnmt.'
1) .lnM»(M+«t)»w»ni(« — «) = — ^^^^,^^^,^,^^^
(l+rioMn(..-)-«))(l+rf-.m(.~«)) ^ 0+.'r4rmV
CO*" am R 1 — k' lin' am II md' am ei
-r coa*aai« '■ ~ ' ' l-^)i'itii*ainuai«*am«(
(l+krinainj[a+«)) (t-t*k*li»ait»(a— c^) (l-t-kt|namuMDCoain«)*
&*ani« 1— k*i
(l_klioa<n(u+«)) (1 — kai»m(u — «)) (l — ki'n
E qiiibiiii..fbnQQlijL etiam sequitiur
eo««m(n+B)coi>ro(n— «) _
1— k'fin'ai
flnn(ti+»^^aiii(n— «t) • J— fc'wg'ain
A'a^tt ' 1— k'l
Posito x=siDsmu, uiBQciscimar' e formula 1):
In am (u -f-w) *!a am (u — «)
1 — k'tm'amei
e fonnulis 2), S) :
1— k'l'^n'aMiti ~ eo.'iin<i. '
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88
e formulis 4), 6): ' '
(l±k»m county _ (l±k5;nam(u+«)) (l±krin«ii.(Q-«))-
Hinc nl)i loco « snccessive ponitor 4 i», 8«», ...2(q — 1)«», loco — • autem 4 n « — «,
olitinemns :
U ITV ~"»iB'»m4«/V "~riB'iin8»/"' A ~ riii'ain»(*-^l)w/
*^ "V (1— k'x»Mii*»m4«) (1 — k'x»Mi»'»m8-) .... (l — k'.*ib'.mA(i) — IJ*) ,
■ A.,H.(u+4.).i..>m(B+8,)...-»m.-(u4-*("-l)»)
f»mcoam*-iiii«»inB».... Ai«)»«n2<n— 1)-}'
(l_k'i'Mp'am4-)(l — kV*in'.ni8»)....(l — k'x'dii*.mtCn — 1)«)
(l+.ia»muUl^-«ll.ln(^H-4-)Ul+«■>•In(i^^^*.)L..-^T+.tn>m(ii+4(B-I)-))
(c«km4«7co«M>A«..'..«MMn'£(B-^4)*>|
m4iii /\ jJD coam 8 « / '
Jtneo*>nC(ti — 1) «
(1 — k'i'rin'ant*-) (l-k'.'iia'»m8*).. .. (l — fc'x"Mn'»in«(n — !)•)
Cl—inimuWl-dnan, (u + 4-)Vl-«n«. ^o+e-)) . . . . (f-*io«n (lH-»(«-l)-))
^ (cM»n4«.«o*w>8«>....co]ainS(a — l)a|'
(l + k»)CC (l+k-){(l + kx.mcMm4.)fl + k.rinco».n8«)....(t + k«wiico.n.I(o_l)«))'
*'* V ~ (I — k'x'.ra'im4»)(l — k'.»mt'.me.)....(l— k'.'iiB'i«.»<«^13-)
f 1+k .in »n «y 1+k Aa am (i.+4 «))(l+k,ri« «in (a+S -) J . . . (l+k >i>> un (u+4 («-!) ■) j
~ {Aan>4-AamB<.... AamS(ii — IJ-I'
/|__^,\DQ (l_ki}f(l — kiiiDcoam4*i)(l — kxuiicoain8»)....(l— ^kxiincoamSCn — I)**)}'
"^ V (I— k'x'fin'»m4-)(l — k'.'rin'am8.)....(l — k'x'.in'.m«(n — 1)»)
(l_k.b.muVl-k«n.m(u4-»«)Vl-k«nani(iH-8»))..../l-kwDa<n(o+4(i>-l)-))
~ ' i[Aam4.&MnSa...Aaiii2(a~.l)aj!
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: sequuBtor £anniihe:
I — u AB /> V lin' coani 4 M/y md' co»in flu/ \ lin' coam > (ii — 1)"/
V ** ''(1— k>i*tm*am4«)(l — k'i'Mi'a«)8«)....(l— k'>'iui*unt(a — 1}«)
cotamn.ui*am(u-f-4a)ca*ani(u+8a)....co*ain^u^4(n— 1)>}
[c<Mam4«.CMain8« .... co*aaiX(a— 1)»)*
_ (l-k'i'*in*coan4.)(l-kVaii/coagn8.)....(l-k'K'iiii*coamX(ii-l}«)
(l-k«.'iui'ani4<.)(l-k'«'wo*«m8i.)....(l-k* «'«!.' am J(o-l)«)
aamuaaiD(u4-4«)Aain(u4-8»)....i&am(u + 4(ii — l)fl)
|Aadi4a&MnaM .... i^unt(it— 1)«]*
DEMONSTRATIO FORMULARUM ANALTTICARUM PRO
TRANSFORMATIONE.
21.
lam demouatremos, posito:
_ (\ riacovain/X Miicoatn8a/ \ iincoamt(n-'-l)M/J
^~ ' (1 — k'i'*in'am4a)(l — k'i'Na'am8«i)....(l — k'>'*iii'am8(n— 1)1.)
(l-Mnau.oUl-*mam(a + 4-)^(l_w.an.(u + 8-)):...(l-«ii.att(u+4(B-l)»)j
lcoaaiii4«.co«am8«.co(>nil2N •■•. co«am8(n--l)sl
et reliquas erui fonnulas, et hanc:
Ay ''_ d»
-f i 1 — \*y' M/ 1 — a* V l-k-i*
sifjaidei
A. = k fHnccwm4a.niicoain8a ....iiiicoamS(ii — I)a|
!nncoain4«.micoaiiiSM (iiicoainS(n — l)i»l
no an 4* . ria ath 8h ... da am 2(d— t) ■}
E formala proposita apporet, miaitneittiilari y, qaoties u abit ia ii-4-4«ii Tcuu awxa
(jniris fector in subse(|aeDtem a}>it, ulUmu* vero.io fu-imum. Unde generaliter y noo mu-
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40 I
tatur, siuuidem loco u pouatur u + 4 pw, tlesigoante p tuimerum intfgrum positivum s.
uegalivuDi. Ul'ivcron=0, fit;
(1 — >;n>m4«)(l— nnam8«)...(l — iiitain4(D — 1>)
[cosain4>.co*Bm8-...c(»ainS{ii~l}«]
'sive y = 0. Facile eoim patet, fore:
— ain ami (n — 1) a ^ ^ sio am 4 e>
— Moain4<ii — SJKi^s+biiiHnSa ,
unde:
(I — ilnan^4o)^l— iiiiam4(n— l)M)=:c«t'ain4a
(1 — unam8i>)(l — ^Ilam4(a_^)■•) = cOl'aID8~
(i-»iuamJCD-l)-)(l-«na.nS(^+I)»)=cot'.,n*C^>-l)-.
lam (iniay=0) quotiesn=:=0, neque mntatar y , nbi loco □ ponitar u -t- 4 p w^ geoeraliter
evanescit j , qaoties n valores indiut :
0, 4>>, 8«, 4(n— S}». 4(a — 1)«,
guibas respondent valores (panlitalis x^sinamw:
0, «a3m4i>, unam8», .. . iliiun4(n— S}*, ■iDain4(n— 1}>,
(juos ita etiam exliil>ere licet:
0, ±>inatn4»i ±3iDam8a>, ■ ■ ■ . i;*inaml(n — 1).«.| ,
sire eliam hune in jnodum :
0. ±*ui*ntS>, ±Mnain4«i, . .. ±*iDain(a — !)■>.
Qui valoKs element! x, quos eranesceute j induere potest, omnes inter se diversi erunt,
eorumqae numerus eril =n. lam ex aeqaatione inter x et y snpposita , e qua profecti sii-
mus, elucet, poaitis:
V = (1— li'»'iiii'«in4»)(l — k'x'»ii.»ain8«)....(l — k'i'iiB'MnS(D-l)«)
= (1 — k'x'.m'an>«(»)(l — k'i'»in'»m4»)....(i— k'i'Hn'ani(n — 1)'»),
yz= — , fieri U functionem elemenfi x rationaleio integram e'' osdiois, ; Quae cuni simiil
cum y eranescat-pro TatsriLas quantitatis x numero n et inter se diversis bequentilnis :
0, ±(iaamt*, j:nDatii4B>, ... ±iinam(D— -l)*,.
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41
necessario fonnam indait: ^ j, ., -' -
M \ Mn*ani4«/l Hii*«ni8i>/ y fin*auit(ii — l)i>/'
ilesfgnaote M ConstaDtem. Com posito x = 1 , fiat 1 — t = 0, y=ii, olttiuemuK ex
U ■■....■..■ ..■:'■ .: .. ; ■. ' - ■
aeqaatiooe 7^— :
\ na'»mtm/\ nn*»mim/' ' ' lio'tm (a' — 1)«/
~ M(l — k'»in'»m««)(l_k'»io'iiii4«^....(l — fc"iin'«in(n_I)»)
( — 1). * (iiiieo*mt*.iiii«<Min4« .... flnci>fm(n— '1}«<J
• 1
f«.iiacoam4«
, |iinfHiS«.Mni«in4« ... jina^tn— ljif| .,;, ,. _ i ' ..';~ '._ .. ^ ..
luter functiones U, V memorabilu interceilit con«latioj^iUu^ itica mptm lu«mor
ratam, cuius iKueficio fit, ut posito ^locq x fipaidf ,in. j-« a)>eat, designante X Con-
staatem.
Posito enim - — loco x abit:
"" ■■ ■'■ ■■ . : . ■ I . ■ . H I . i ;
in hanc expres-'tionem : ''■ '
£^ V 1 ^
Ml" k°(*iiiainSa.>ValLin4a'..... ad am [n — !)«)*'
Coutra Tero eadem snbstitutione facta , . . - , , ^
in hanc expre-ssionem a]ii,t; ■ . , r .: . .;.,,' ,:..-.;
{— 1)~^ — ^ . M {iiD amto . HW.B.4* . . . .-*i»«n^ — IJ*}..
F
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4i
Undu loco X posilo-r-, y=— abit in:
sire y in — , slqnidem ponitur:
X = HHk'^{«in*mXa.*inMn4«....Mn>in(a — 1}.}*
= k^luneoamSa.HncaaiDAa .... iinc(Min{n — !)■}*
Id (juod demoDStrandnm «rat.
Ex aequatione propositn:
(l — V*»'mii*«iii4w^(1 — k*i'Hn*am8M^,...^l^k*x*sin*ainC(ii^l)«)
posito -r- locox, — locoy, qnod ex antecedeQlibus licet, eruimus: '
_L_ 1 3= -i^^|(l— kxrin(i«m4«)(l_k«iineo»fh8«),...(l — i<MocoMiit(B— l)-)l'.
qaod dnctiMii in Xy = -^, praebet:
l-Xy = (l-k,)ii ii V ^ ^ ■ -^■
Ceterumpatet, y=~al)ireiii — y, ubixin — x mntatar, qao fecto igttor statim etiam
l_l_y, iH-iyexl— y» 1— Xy obtinemos.
lam igitor eiiumodi iaveiutiuu flinctioiieft dementi x rationales integras V, V,
ut sit:
V + U= V(l + y) = (! + .) AA
V ^ r ^ y (1 - 7) = (i — .) BB
V + XU= V(l + Xy)=a+k.)CC
V_XU= VCl — Xj)=(l-k>>DD,
desigmiDlibiis A, B, C, D et ipsis fnnctiones element! x ratiouales iotegras. Hinc aatem
secuudam Frincipia TransfermatioDis initio stabilita statim seqaitor:
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Mahiplicatorait M, <jueni Tocabimus, ex obserratioae §. 15 fiicta obtinemiu. Uiuk
iam omnes fbmiDkie aiialylicae gener&les, quae theoriam traDsformationis fuocliooam el-
lipticamin concemnDt, demoustratae sunt.
22.
Demoiutliutiio proposita ex ea, qaam dedimas in TSov'u Aairouomicis a CI. ^ha-
macher editisNo. 127, eruilur, nhi poaitur w loco — aliis omQiljiu immutatis maDenti-
bos. Ipmin theorems analyticnm genorale de TnmsformatioDe sub {bnaa patJo alia iam
pritu ibidem Ifo. 128 cuni AualysUs oommuuioaTcnun. Damoiulralioiiem CI. Legendre,
»ummus in faac do^triua arbiter, ibidem No. ISO benigiie et praedare recetuere voluit.
Obserrat ibi Vir malli« nominibos TeDerandus, aeqaationem:
V^_U~ ABCD _ T
cuius beneficio demonstralio conficitur, et qnae nobis e principiis transformatiouis mete
algebraicia seqaebator, etiam sine illis atiatytice probaii posse. Qaod com ex ipsa Viri
Chuusimisententia egregiam th^oremali nostro lucem afiundat, praeennle illo , paucishunc
in raodum demonstrenius.
Aeqnationem propositam:
v-iH._u— *BCD T_
di , ai ^ M " M
ita qnoqne exbibere Hcet:
dU dV dIogU dIogV ABCD _ T
Udi~*"vdr™ di di ^ MUV ~ MUV ■
Invenimus antem:
^ ~'mV~ MB*»mt-/r~ .iii'.n.4« j--"!^*— .in'«iii(o— J)- j
V = (l_k*»'.)«'»in»-)(l — lL'x'«m'am4-)...(l — k'i*«"'«»(n— 1)»).
ande:
JIobU dlogV _ 1 ^( ^
1— k'i*.u>*»tt4- f
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T ABCD
.lUV ™' MUV
namero q in sunima i?esigiiate (riliulis-valorifius r, 2 , S , '. . . ■, ■ "Z' " ■ ' Tom inveninius:
CD = (1 — V'i'siii'coain2«»)(l — L*K*»in'coain4«i)....(l — V'i'»iii*c«»mCn — 1)«(),
nnde;
Yl(i " Ul_k'x'«i.'co.in2pi.)
slquidem Id producUs breritatu. caasa pifiefuEO sigoo n denotatis sbmeuto p T«lcres tri-
bouDtiir l,Cr 5} .-..., ~7~ '* ^'>*' «x|veS8ionezn iu frafetiones aimpUces discerpeK
iicet, ita ut formam ibdaat: " •
1 / a"'» ' ' b'^'x
quo facto at evictam habeamas, quod propositura est, denioostrari debet, fore;
Deaolaliimos lo sequenlibtu praefiso signo n*^' prodtictum ita formatunij ut ele-
meoto p valores tribuantor 1 , 2, S, ...» — s— > omiaso tamen valore p^q. Hinc e
praeceptis fractlonum simplicium theoriae abupde gratis 9e<|uitDr:
«in'»mgq»
'^V l_k*>in'iii>SqM.siQ'aniSu«i /
= jl — k*Mii'sni2q Ai.*iD*co*ni2i]a>l —
*A ^ 1 — k'lia'amXqai.ND'coamZpw /
lam e formulis supra a noljis exhibitis Rt:
^ ' lin'imitqai
T Mn*ca>m<p0 co(>in(Xq+Sp)«.eoi«m(tq — Cp)"
1 — k'ua'amtq oiin'aDiZpN co«'am9ptii
sin' am 2 q u
rio'trt tpm _.^ coj coam {Sp+4 q)a» . botcoam (Zp -i-S q) u
mSpn coi'ccaraSpw
Digitized b, Google
,48
FacUe antem patet, sublalis (jai iadenomipatore el uumeratore iidem iareoiaotur facto-
rOiaa, fieri:
*' cM*>in2p« ~~ t^tfatq*
unde :
.(q) — (1 — k'MD'«ii»tq>niii'coani2q*»)c o»e)win4qM
cu an f I) *i coi coam 2 ^ u
At e oola de <luplicatione formula fit:
2 k uoam 2q«tcotani2aatAain2q •>
>icoain4qai ^ - i
1— 2k'iiii*«in2qBi + k*Mii*am2q«
2qBi+k*Mii*am2q«
2k •iniin2q«coiam2qu^iin2qai
A'ainXqaf — L*«in*aD)2qwcw'«inlqM
2co«>w2^«.«— coawigqM
1 — k* *iD* ■■i2 q M fin* coim tqm
onde laudem, qaod demoiutraudum erat, A^^'= — 2. Pnwstu simili modo alteram oequa-
lioDem: B'^=2k*aiD^am2(j w probare licet; (jaod tameo, iam ioTento A'^'=3 — 2, fa-
ciliufl ita fit.
faaie patet, loco x po.sito -r- non mutari expressioneia :
nCl ____!__ Ul_k',',in'co.m2p«|
V «n'coa.n2p«;\ '^ }
qoatn vidimus aequalem poni posse expresiiooi:
.+2_^'J= + 2 li^Ji! .
fin' am 2 q M — k' 1 — k' »ili' am 2 q w i'
Haec aotem expres^, posito -r- loco x^ abit is Iwuo: . _. r
* t — k'a'*ii>*»in2qM k'(»ili'»m2q u — i*} ~~
^ T (9 °'^' ^ y 2fc'i'»in'amtq*. y — b'^' »'
''*■*■ i k' MO* am «q •/■*'■*• 1 — k».»«B*«oi2q» "'"'*' k'rin'aDiZqu' »ii'aiiiSqa> — x' "
.unde Qt immntata ilia maoeat, quod debet, fieri oportet:
B''*=2k'»in'am2qa..
0. D. E.
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■46
23. •
E fonniila 14) §. 20 se<jULlur:
n7?^i/TF? ^° ^ /Yp? ^^~''''''^'"^*''X*-'''''"°'«^*'')---(t -^'-'"°'''^("-^)_«')
(l-k>i*Mii*«iiitt>Xl-k*i*nii*am4u)...-(l-k'K'*in>aai(i>-l}»)
Fositox=lj unde eliamy^l, ac v 1 — \X=:X\ fit:
™ I AamXM AaaliN ... Aam(n — 1)m /'
lam vero est:
k'
~ A IDI U '
uiide:
1) x' =
k'°
{&am2ai.&ani4ai .... a«n)(n— l)uj*
Forro in asnm vocatia fiumolia :
tj X = k"|Miieo»tn*i>i.»iiicoMn4» .... rineoMn{n— 1)»
8) M=(-I)
nancUciniDr :
1 — * |tinccaniXw.iiiica*in4w .... ibconn(n — l)a(|'
o.«w..iDim4w....«nim{n_1^7j
{coi»nX«co«am4w .... aMain(a--t)tt
Maiiil»^am4w .... Aan(D— I) wt
7) ^ 1/ -;^ = Jtg wn « w . lB«in4«i . . . tgain(i»— 1) «.|
lin coam Z at . lia coatn 4 ai ■ ■ ■ nn coam (n — 1) mJ
jcoi coam S ti co* coam 4 n
db, Google
10)
n— 1 r~ / >j
11} (— 1}"^ H m/ . ^ ^^ mc {lg coam t M Ig ci»m 4 a . , . tg cram (n— 1} m\ .
Harum formulanun ope formulae l), 4), 5) iu sequentes abeunt:
IX) «nam^-^, xj = y ^»iiiaiu«B«o.(u+4«)«tiain(o+8«).. .. »i«Mi. («i+4{ii— 1)»)
IS) coiw/-^. x) = y i-!^«o.Miiicoiiin(tt+4«)cwuo(u+8«.)....eraam(iH-*(n-l)")
14) &Mm ^^. >.j=y ^iM.ua<n.(u+4«)a«n(u+8w)....aai«(u+4{ii-t)<.).
node etiam:
15) »g«n(-^. xj = y iltgtmutg«n(a+4»)lg«n(n+a«)....tg.m(«+4(«-l)«).
AHud ita inveiutiir formularam sjslema. Ex ae<jaatioDe 4} seq^oitur:
= {lin im X u tin am 4 u . . . . siii am (n— I) wj ,
uiide:
/« .\ » TT •in'.mSp*, kM Tf
= ««am(_. XJ=-||.j-j_-j--_-=_^,||-
— uD'amXpu
k'lin'amSpM
°—nc--i.--»'p-)-t^.H..»(-^.>)n(.-- t.^.L.pJ -
Kadices haiiu aequatiotus u" ordinis sant:
i = iiaaRi«. Niiani(«+4a(). rin am (m+8ai), ..... tin am (u+4 (n — 1) w).
nude aequationem naDcisciiniir identicaui:
-n('-^»--»-)-nr-'-(^.^)n(--p;5:|^)=
,, Google
4$
Hinc prodit samma radicam
16) 2 *•■" »■» C''+*q") = -nr "" "" ("^ • ^) ■
Eodem modo invenitnr:
17) 2 <'■>"■" ("+*'»''>= hM """(la- '^j
19) 2 >e »- {«+4q~) = t^ »B •■" (-S"' *■) ■
in quibua formulis uumero q trUiuuBtQr ralores 0, 1, 2, 8, ... nwi. ^uas fonndas
etiam hunc in moclum repraesenlare conrenit:
kM VM / V , ■ ".
D— 1
(-1) - >•.- (^ ^1 = M.«n« + 2!^»"'("+*'I'')+'**""("-*'«-5l
kM V™ / ' '
— j^ Ig «« (■^. A^j = tg -m a + 2 {le^"- (•"+*q «) + t«.™ (»-*q«)},
ubi numero q tribuuntur valores 1,2, 8 , • ■ - ^y~ * '"'" 8*'^'*«'**"' fonnOlse :
S eo» »iD 4 q <• i »in 4 q w »in iw ii
5m*mCu + 4q«) + im»m(ii-4q«)=
coj»in(u+4q<u)+co.«m<u— 4q«,) =
A am (n + 4 q o») + A am {u — 4 q w) =
tgam<a+4qo.) + tB.tn{u— 4qu) =
l-k'.to
>m4q«
lin' am a
X cot am 4 q « CO*
jm u
l_k' sin
>.n.4q».
n'am u
XA»
m4qaiA
itnu
l-k'«D
am4q*,
2 A am 4 ^
a un am n
cof am u
CO*' am 4 q ai
_ i' am 4
q »• ""' »"• «
•) cf. J. 18 formuta* I), 1^. 5); formtja pJslrema e' formalU.K^, 3<^ Huil. uW repufci, cue Igff + 'e''
lin (g + J)
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40
(|uarara ope formulae 16) — 19) in has atrennt:
») — J— ; — niiMn|-r^> ^1 = *m im u + 2 — ; — 0-:^ ; ~~z
kM \M / 1 — k*«in an 4q m na* ain n
— '**'*I"M • »-| = e««" + Z — ; — rr-ri — ^ tt
\ Ja. / ^ 1 — k «in •» 4 q w «d* am u
81)
i=lLi:A..YjL. .^ . ■■ .. ■ 1=. «A.-4q.^.m,
1 — k*uii>«m4
t a am 4 q w »in
«) ^^^^^ ' ■- (^. *) = a .» " + 2 i_i..^„..„,,|. -^
(juae eliam bhtinentur, ulii fbrmolae sapra propositae e methmlix notis in fractiones
simpHces resolrantar.
DE VARUS EIUSDEH ORDINIS TRANSFORM ATIOMBVS.
TRANSFORHATIONES DUAE REALES, MAIORIS MODULI IN MINOREM
£T MINORIS IN MAIOREH.
24.
sigiintitiLus m, ni' numeros integros positivos s. oegativoii, quitnmiiq, quories n est ou-
merus coinpnsitus, nullntn ipsius u facrorem cotnmuuem habeu). Facile autem patet, nl«t
4 sit primus ad u, valores w = -^ — ^^ ' — suhatitutiooes diTcrsas uqii e?chiliituro)i
esse. Hinc ubi ipse n est nnmerus primus, valores Klemenli a*, (jui tranvformationes Hi-
Tersax suppeditant, erant omiieft:
K_ i^ It+iK' K + t i K' K + gJK" K. + (a_l)iK'
ii'd' u' n ' n .•••^
sire etiam:
^ IE. ^ + . ' ^' « K + i K' 8 K ■!■ I K' (n — I) K + i KT
aut, si placet:
G
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Google
quorum est numerus u+ !• Ac reapse vidiiuuit, , in U'aasformatiombus tcrtii et quioti or-
diub, SDpra tamqaam exemplis propositi^, ae(]uatioae$ later u = ^k et r = y^X, qiias
Aequationea Modularet Quacupabltnus , re.sp. ad rja^rtuni et sexfum gradum asceudisw.
(.)uoties vero n est ntunerus compositas, isle valde augelur uumrrus; accedunt enim ta-
sas, qui)ms sive m, sive m' sive etiam ulerque &cU>rem Ifaliet ci|m n couiwubmq , modo |
ne uEris<jue m, m' idem comiuunis sit cum u. Geiieraliler aulem valet theorema: I
„ numerum aubatiUUionem n" ordinia inter ee diveraarum , quorum ope tratufor- I
„ntare Uceat functioriea elUptictu, aequare aummani factorum ipaiut n, qui ta-
y^men numerua, quotiea nper quadratum dividitur, et aubatiautioaea ampUctitur
„ex tranaformationa at midliplicatione mixtaa; adeoque quotiea n ipaum est qua- '
„dratum ipaam multiplicationem." i
Lta igitur factorum summa desi^nahit gradum, ad (|uem pro dato namerq u Aequatlo Mo- ^
dularis ascendet, nbi adDOtaudum est, quoties n sit Humerus quedratus, uoam e radicam
numero praebiluram esse k = X, ac generaliter, quoties u = m'v, designaule m^ qoa-
dratum minimum, per quod uiimerum n dividere licet, e namero radicum fore etiam omnes
radices Aeqaatiouui Modularis, quae ad ipsum v pertinet.
Inter valoreselementi w supra propositos, quicasa, quo n est primus, quem, cum
in eum reliqui redeant, sire uiiice .sive prae ceteris considerare couvenit, universam traiu-
formalionum copiam saggeruot, duo tantum generaliler loquendo *) iiiveniuntor, tiui trans-
formationes realea suppeditaal , hos dico w = — , w =: . Ulam iu sequentibus voca-
himus transformatioiiemfrimam, haac aecundam ; modulosque qui his respondent, desigoa-
bimus resp. per X, X^ eorumque Complementa per X', i^. Argamenta amptitndinis — ,
quae his modulis respondent, (functiones integras vocat CL Legendre,) desigualitmn.i per
A, A,, a', a/. Formulae noslrae generates pro his casibus evaduat sequentes.
•) Nam infinili) cuibua pro ModulLi ipccialibus fit, ul fu ladicam imagiiuriiniin Aeqnstiooum MfMluUrimn
ribi Miotic evadal ideoque rule fil.
I, Google
61
I.
FORMULAE PHO TRAMSFOBMATIONE REALl POI^IA MODULI I IN MODULUM X.
X = k"
tK . 4K . (ti — 1)KJ
lAani dam ...... a am ' — J
Inn n t
I . «K . . 4K (o— I)K i'
I iiD coani — — MD com) ■' nn coam i- •■ — — I
~ \ tK . Tk " iTn — llK '1
f tin am -— «m aoi — md ani I
\ • ■ . " )
M I ., «kU'-., «k|-|'-T^ (.-1)K|
,ji_ X V "■•"T/\ •■"••■»-; \ ..■■..— 1^;
""■"^" ■ ■" (.-k-.i...,„!iaio.a„.j(.-l-d.-.„*Ji.f„'a„,.j..,.(l_V.i...„<:ri«£.,...„„j
,—- Aamuf !->'*■■> Oain — aia'atnuH 1-k aurcoam — ain amu I ...I l-li'fiu'c«aiii — am* am n I
"'«■ ^"V i-- (,.l,,,...„»^..,„.,„„)(,.l.ai...„^.i.-a.,.j....(.-l..i..,„!llii5.i„.„~
( .i«»nu 1/ aiaai.. v /. ■h»"° \
' ■ ■ • I tincoam — If iincoam — I 1 tiacoam-J^ 1— I
/
V
.X)
,.)
8K .
eoam — ill
u
.„„)
.■(.-k»..co>m«<"-'J'^^n
1— kiioaffiB
»")
H4«n.»o
(,+k.aco.m^.taa
""X-+'-
co.m**'.;
n am u 1 .
..(l+k»i
t(n_lJK .
oeoam-i i— lip
D
Digil zed ;
, Google
-ra-™""(-H-' ^)-=v—' •+«2
(~i)^MfmZ3 _i,|
-kM-"""(T' »)=«■" ■"•+«2 . ^. . . t,°K . .
-^r^-'i-w- •)-"""' + '^-
-ina-'f°(-^.>j = f"- + «S ,; ,K..
^ ' co^ am -^ A' am --^ nn' ui a
^. FORMULAE PBO TRAHSFORHATIONE REALI SECUHDA. MODULI k IN
HODDLUM X,. sup FORMA IMAGINARIA.
fiK' . 4iK'
n'
-I
M, -(-!)-
"(i-.)-
tJK' . " Til? ; (m— QjR' f
'-tt~Tk|('— — fTS')-|'-~nEESIEI
y ^.iiniiniiiniiiii(ii-|-4iK')iia.m(a-|->IK')....>ii>iii(ii+«(it-l)IK')
Digitized b, Google
I .. ■•KU ., JiK'l I' ., (._«)iK'l
" V J^'»'"'"=""»("+—j"-°(°+— )•■•■"— (■+ — ; — )
"" \ir • \\ = !-i —, —t
'•"■ I /, "°"°' \/i ■ ■"■•°" \ /i "'"" \
\ "■■••■»— /\ w.m— ; ^ ...,„___;
/ '-■°"(-ii:'4-. .
. %\v:\\~, *'*t'I "r"^ . {— i)iK'|
n / \ n / ^ n /
-fiioMra /. . MD»inii \ /, L "" '" " \ j, . »iii»mii ^
7?
■(-^•^)
„(^...)
1 iin coam II un coam —— I 1 nn cosm I
CMain'^~^! i iam— J nna
WM- "■ - (■j^ = » "■ • - X 2 ' .. '' it^-DIK- .' .
' ^ ' t ' md' am ■1-' ^ (- fro* am u
d by Google
( — 1)^ A am lio am u coi am u
B. FORMULAE PRO TRANStORRtATlb^E REAL! SECtJNDA SUB FORMA REALI.
I— 'H^' , i.)i
('^..»(i,.))( t?;;(i^,..)|; (■\™(i^..j)
""Im"' M^ - > -1 . r. sin' Bm u- \ /, . . "n" "m rf ' \ /, . ( wn" am b \
■ ..^.('*,,.(^,.))|%..„(ff,.)) (\,.(t^4
Digitized b, Google
/
(1 — vnamu^amf •^')|| 1 — tba^iuAaml — , k'l I...I t— tMi»"<<'^*ii'( • ^'l I
('-H-T1^■-,("■.'■))(•4.■.St'>T^|^|■)).■•,^('+"•r:■'^r(T^•'■))
'..-.1 in) ,
, Y>.-,v»!""(Tr'^) .
' .i«-,„fJ5l!ii;.k-)+c«,.™(!<3^'.l-).i.-.n,«
(-«.™..„r!!l:i)5:.i.U„,.
kM,
'00*
-fe
.^)=.
B-l
A *
"■{w,
-•)=
-A_^«.(-^.v) = ^«.u + t2-
I— i , k l + cOi iml — i ,k Isin'u
,, Google
— - « —
« — t
In fbrmuUs pro trsnsfbnnatione prima ponittta est ( — l) ' M loco M. Formulas
pro tranitforniatioDe aecuuda dapUciter exEilwre plapoit, et sub ftirma imagioaria et suIj
fonna reali, in (jDibus praeterea loco k sin am , k sin coam —^ — , cet. ubiqae
MTtptum est f—i-rrs^'t " ; — . .. - i cet. td quod, sicuti rednclio in
tio ini rin conn -i -^-^ —
I'onnam realem, ope forraQlarnm ^' 19 Tacile ijnutMtctuni est. Ubi signam ambiguum +
posilum est, alteram + eligendum est, abi — % - Mt oumerus par, alteram — , ubi — ^
est numenis impar; de signo'^ cootrarium valet. In stiminlis praefixo £ designalis, nu-
mero q valores 1 , 2 , S , .... - \ ~- trlbnendi sant.
B Jbrmults pro trandbnaatioBe prima proposilis paiet, quotios u fiat succes-
tive:
o JL — — *^ .
fore am (^.x):
0. _. „. _. ,..
node obtinetnns :
Contra Tero viilema^ in Iransfbrmatioue secunda, quoties u fiat: o, K, 2K, sK, ...
sire am b: 0, -j-, ft, —-, . . ., 'fieri am (-j^. >■,) et ipsam=0, -j-y ^> -r- 1 • ■»
nude hoc casu:
Ceteram e formulis pro Modulis X, V, X, X' exhibtlis elucet, crescente n, Moda-
los X, X^ rapide ad nihtlum courergere, ideoque sitnul Modules X', X, proxime accedere ad
uaitatem. Itaque transformatioDem Modali primnm dicere convenit maiorit in minorem,
secundam nUnorit in mtUorem,
db, Google
6T
DB TRANSFORUATIOHIBDS COHFLEMENTARIIS
S. QUOMODO E TRANSFORUATIONE MODULI IN HOODLUM ALIA
DERIVATUR COMFLEMENTI IN COHFLEMENTUM.
25.
In fonnula supra inTenla;
'••"('H'' '•)-V-7(r-'« •»"'!■"(" + ••)'«■"(" + '")•■••'« •"("+•<■-')•)
ponamus usaia, «=siw% in at Mt <f=inKT4-ni'iK', o'sm'K' — miK. lam vero
est (}. 19)
ll.»(iu'. X) = iii«.m(«', V).
unde formulam allegatam in aequentom abire ridemus:
Porro iaTeoimiu formulae;
»— ' fNBcMmtvNniMui4a...*iiicoBin (n— l)«t*
|»i •I. taiin^m <•>... ilii an (i— 1] •{=
(jnae e fomudis:
Aia(ta, k)
fio cokin (u, k)
lode etiam le^itar:
-i_WIC^«(«,k')
linamOu, k} ig m (<• . b'} & >m (a . k*) iinun(u, k;
I aeqaeoteft abennt ;
K' = k'»{«ii«»ni»«.'iuico«ni 4 ».'.... lin CMin (n — I) •>■]* (Hod k')
M = — ^-: r-r-J r"7 r— tt — ., ,.w ^ I"*^ ^ 1
h' lin an %«' ..'. (iB an (n — 1} wj
H
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58
His formulis oomparatis cum illisj quae traosformationi Moduli k in Modnlum X ia-
nerriunt:
Hnaml-^, X J = ••/ --— nnain n mq mm (u+4«*) mo •« (u+8 •») Mn an (u+4 (n— 1)«)
X = k"(«DM»int wrincoimi* .... lin coini{n— 1) mJ*
°~* f lin eoain 2 w lin coam 4 u ,, , . lin coam (n — 1) n i*-
"~'~*^ ' \ >ia im t •> NB am 4 N ■ia«ai(n — 1}m /'
eiucet Tlicorema, tjuod tnaximi momeoti censeri debet in Theorta TransformatiouLii :
j^Qudecunque de Tranaformatione Moduli k in Modidum X proponi possint
„ formulae, eaedam valere, mutato k in )l\ X tn X', tt in tt =s -:- ^ M ui
o — 1
»(-l) ' M."
TraDsformationem autem Complementi in Complementam, dicto modo e transfomiatione
propouta ' derivatain , dicemus TfaatformationMm CampUmenttineaif.
Facile patet, traDaformadonuai realium Moduli k tranaformationes reales Moduli V
complementarias esse , ita tauten ut primae Moduli k secuuda Moduli k', secondae Moduli
k prima Moduli k' complementaria sit. Ubi enim in tbeoremate modo proposito ponitor
w=:-^ — , 1*=:-=^ — , quod transformationibns Moduli k primae et secandae respondet,
htm'=-^=:— — , «'=:-?-^ , quod transformaliooibus Moduli k' respondel reap.
Mcnudae et primae. Nee non, cum crcsceote Modulo decrescat GunpleueDtum ac vice
versi, trans&rmatio Moduli in Modulum ubi est maioris in minorem, traosformatio Gom-
plemeuti in Complementnm sen translbnnatio complementaria minoris in maiorem esse de-
bet, ac rice versS. Videmus igitor, mutato k in k', abire X in X^, X, ia. X'. Nee nm
Mnltiplicalor M, transfarmationi primae eiustjue complementariae communis*), abibit
*) Hoc pacralitcr tantiim ncglecto Hgno Taletj Tidimni ciiim, qnod in alkra tr. wat M, in eomplanenUri*
cue ( — 1) * M; It noitris catibu* to, qiiod in Imuformationa prima loco H poMfooi Mt (— I) ' M
(v. lupra), ligni ambiguiUf lolliliiT', ita ut traniformatioiiibua realibiu complanenlariit' oamino idtm lilUn]-
ti^tcatoT M.
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TRANSFORM g^ SUPPLEMENTARIAE. c5 ««)
"Inda cum supplkmentabia.
■ ■ I ■--.■
H t
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68
Hia formulis oomporatu cum iUis, quae traosfonnatiom Moduli k in Moduliua X in-
Mrriont:
«D.iii(^. \) = y -^ihiaio»«ii«n(u+4«.)iiDmci(u+8»).....iiiam(u+*Cii-l)-;»
A. = k^funcoamf Muncaainlw .... nncoain(a— 1} »»J*
£^4| *iD coitn 2 01 (ID coam 4 u ... . tin coam (n — 1) oi 1*'
( *■■* ■■» S u ria am 4 w . . . . uo am (n — 1) w / '
elucet Tli(>orema, quod maximi moinenti censeri debet inTbeoria Transformaliouiii :
,fQudecunqu€ de Tranaformatione Modali k in Modulum X proponi poM
„ formulae t taadam valerg, mutato k tn L', X in X', w t/i w'= -p, M
\
n — 1
Transformationem autem ComplemeDti in Complementum, dicto modo e tratufonnatioii
propouta ' deriTatain , dicemos Tf'aiuformatioatm Comptementariaitf.
Facile patet, traDsformatioDam realium Moduli k transformatioDes reales Moduli J
compIemeDtarias esse, ita tatnen at prlmae Moduli k secnoda Moduli k\ secundae Modu
k pfima Moduli k' complemeutaria sit. Vbi eoim in tbeoremate modo proposito ponitD
t,-=— — , u=i-^ — , quod transformationibas Moduli k primae et secundae responde
fit w'=:-^=:-=^ — , m' = -^^- , quod transfbrmatioaibos Moduli k' reapondet respl
Mcundae et primae. Nee uon, cum crescente Modulo decrescat Gom^dsBientam ac vicA
yersh, trensfbrmatio Moduli in Modulum ubi est maioris in minorem, transformatio Gnu-I
plementi in Complementum seu transformatio complementaria minoris in maiorem esse de-\
bet, ac vice versfi. Yidemns igitor, mutato k in k', abire X iu X), X, in X'. Nee ikhi
Multiplicalor M, trausftfrmationi primae eiuM|ue complementariae commaiiis*), abibit
*) H«c ga«raGt«r tanhnn Mglecto n«uo Taletj vUinua enim, quoi in altera tr. aril H, id eoinpl«Mcnlat»
cue (— 1) » M; at noftrii caiibn* to, qiiod in tmufonnaliona prima loco H poiifiun ail (— 1 j~* ' H
(v. tupra], tigiii ambiguhw (Dllitur', ila ut trauibrnuliambiu realibiu complcmenUriu- oamino idem ril MpJ>
tipBcator M,
Digitized by GoO^^jle
TRANSFOm, g^ SUPPLEMENTARIAE. (5 aso
NDA CUM SUPPLEMENTARIA.
■(-' ■^,; ,- .1
H I
, Google
,, Google
in M,, qui ad tnnsfonnalioiieni seciuidain einsque complemeDiaruie pertioet, ac rice renfi
M, idM. Hioce formulu snpra inrentis:
seqannttir hae:
unde proreuiaDt fonnulBe stutuni momenti in bac theoria:
Hae fbrmalae geouiDiitn tTamfonnationis propositae cbaracterem coiutituunt, unde palet,
bono iuie singulas nos transformatioDes ad singnlos nnmerOH n retulisse. Adnotalio , quo-
ties Q sit Dumenu compositns = n'n"} e singulis radicibus realibus AequatioQum Modula-
riDm, seu e singulis Modnlis realiLus^ ia quos datum Modnlnm k per aubstitntionem a"
ordiDLs transfbrmare liceat, proTenire aequadooes buiusmodi:
quae siugulia discerptiooibos noia^ n in duos factores respondent. E quanun igitur nu-
mero, quoties n est nomems qaadratns, erit etiam haec:
quae docet, casn quo n est quadratum, e nnmero substitatitmum esse unam, quae mul-
tiplicatioDeni suppeditet.
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DE TRANSFORMATIONIBUS SUPPLEMENTARIIS AD
MnLTIFLICATIOSEM.
26.
Berocemiu formulas:
A' _ K' a; l^ K'
A^^It" A, "oK"
f|uil>iu haoc in modom scriptis:
elucet, eodeni modo pendtrt ModuUtm X a Modulo k atque Modulam k a Modulo X , titv
eodem modo pendere Modulum k a Modulo X atque Modulum X^ a Modulo k. Itaqiie pa i
transformatioQem priraam s. maioris in miuorem, qua k io X, transrormabilnr X, in k; '
per transfonnationetn secundam sen minoris in maiorem, qua k io X,, transfonnahilur |
X in k. Itaque pott tra/ufonnationmn primam adhibita wcttnda aeu poat teciuidam adAihita
prima, Modulut k m «« redit, am* trofuformationui prima et aecunda auccetaive adhibitat,
utro ordine placet, Muitiplicationem praAent.
Vocemns M' Maltiplicatorptn , qui eotkm modo a X pendet atque M, a k ; M^ Mnt-
tiplicatorem qui eodem modo a X, pendet al(|ue M a k ; ita ut olttineaotur aequatiooes :
*7
Aj
quamm alien transformalioni Moduli k in Modulum X |>er transfbrmalionem primam, al-
tera transformationi Moduli X in Modulum k per Iransformalionem Mcandam respondft.
Kx his aequationilins proveoit:
A*. dx / u \
Vd— Oa-k'**) MMVa— .•){l-k''') VMM'/
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At ex aetjoalioiio A, sr: -j^ inalftado k in X, quo /acto K to A, X, ia k, A, io K » M, in M'
aliit, obtioetiir K=-r-7-, qua aeqaatioue contparata cam ilia A= — ^, proreiiit
Eodem modo ex aefjaatione A = -~t7- mataodokin X,, <|uoiaclo K iu A,, Xin k, A in K,
M, in M^ ahit , provenit K = — =^ , qua aeqcatioue comparata cum hac A,= -rj- , proveiiit
1^^=:n; nnde ridemos, ddobus illis casilias ponit Maas transfonhationes mcceftsire ad-
hil)itas multiplicari Argameottim per narnemm ». *
Ufii past IraosfonnatioQeia Moilirii k in Modulam X Modolas X rarms in Modulum k
transformalur, ^ nt Mnlliplicatio prorenial, Iianc transformaliotiem illiu» aupplementariam
ad muUipliceaiontm MQ &imp)iciter aapplemsntariam nnncupabinnis.'
AppoDamus cam exempli causa tarn ia.usuoi se(|uealium formulas pro transforma-
\iaub primae tuppUmeataria, 9. Modali X in Modulum k , <|u«>.erit ipeiosX secunda, eaH
tameu sab altera tanlum forma imaginaria, cam reductio ad realem in promlu sit. Quat
confestim olrtiBeflMH fimnulas, nbi in iiit, quae supra de trattsfornyitione Moduli k seconda
propositae sunt, (t. tab. 11. A. $. 24) loco k poiiimus X, k loco X, -r^ loco a, M' = — - ^
loGoM, node -jg^ = DU loco -^. In his formalin , sed iu his tantam, Modulus X vale-
bit, nisi diserte adiectos sitModnlnsk; ceterom brevitatis causa positumj=sin ami— , M;
nDmero q , ut supra , tribuendi sunt valores : 1 , S , 8 , ■ > ■ , — j — ■ —
dbyGoo^^jle
rORHULAE rRO TRANSFORHATIONE MODULI X IN MODIILUM k,
SEV FRIUAE StirFLEHENTARIA.
27.
. ,„ / . «iA' . tiX (n-lJiA'i"
k IB X <iin caim ^— tin coam . ■ . . am eoaiii i
( n n n /
I . tiA' , 4iA' . <ii— l]iA' 1 *
tiA' . 4iA'
. (n-l)iA'
A M (ma, k} =
/X" . u . /u . *iA'v . /u 8iA'\ /u 4Cd— I)iA'v
, /i_ iz — \/i n — ,\.../i u \
y ...'—^^/^ .n. .»-^; \ «■.'■« — ; — ;
V^ t'X' « /ii 4iA'i /" ."A'l /« «(»_l)iA'i
n?r"-""-H-""»(-M+-r)"""(M+— J-"""{m-+ — J — )
Digitized b, Google
/
, . , , , /". \ MB co»m /I on cokin f I mi coam I
1 — »m>in(i»ii. I) / 1 — y \ n /\ a / \ n 7
/
I MB eoam — — 1 1 tia com — — I 1 an cava ■■■-' f
. Hs-DiA- „ ._ (I,-1).A-
^y *T ^
knM knH *
.,...„ ii—i „
(-ir[v/v-j7 j/T
^ ikoM •*
(-l)«™.m
gl-')'A'.._i«<-lliA-
.>*
.„<... k,=t^A357H+i^^SS^:
Ig Ml (n n , k) s
.-.
'«.^
..„A' ^^
(-«'
nnai
^(g<|-')
n
ii^»..«^
■Si
iA'
HD<
.„2l
..,iA ^^
' llL^_A' *S'^' ■ .
Theorema anal/ticum generale, tntoiformatioDem illam primae supplemeotariam
coucerneDs, iam initio tnensis August! a. 1827 ciim CI. Legeodre commuoicavi, cuitu
etiam itle jn N«ta supra citata (Nora Astr. a. 1827. no. 180) meodoikevi iniicere Toluit
Simile formularam sj^stema pro traosformatioDe altera secundae sapplementaria s. transform*
madooe Moduli X, id Modalum k stabilin potuisset. Quae omnia at diluci<Uora fiant, ad-
iecta tabula formulas faodamentales pro transfiDrmatioDibus prima et secunda earam com-
plementartis et snpplemeDlariis conKpecttai expoiiere placuit.
d by Google
64
Nee uou e DDmero traosformatlotiuiii imaginariaraQi una (juaeque «aam baJfet ao^
plenieatariam ad MultipIicDliouem. Sup|KinBiuuit , tfuod licet, bumeros m, m' §. 20 feclo-
rem communeia non habere: sit porro m/u' — ^m:=:i, designaatiltas ^ , ^' nnmenM inte-
gros positiros s. oegativos. lam si in forniulia noatrifi generalibiu de traDafonnatiooe pro-
posltLs §. SO sqq. ponitur w = '" Jt ' — , ac k et X inter se commatautar, formulas ob-
lines, (juae ad suppleoientariam lraiuformatioiu» pertiueiit. Fotulo m= t , m'=;0, fit
^=0, ^'=1^ oude ^ ■ ^ - ' — = -^|j-=: -— -, quod prinne aupplemeotariam prae-
bet, Qti vidimus.
FORMULAE ANALYTICAE GENEBALES PRO MPLTIPLICATIOKE
FUKCTIONUM ELLIPTICARUM.
28.
E binis Trausformatiooibus Sapplementariis compoaere licet ipsas pro MultipU-
cadone formulas, s. formulas, qnilius fonctiooes ellipticae Argnmeuli'mu per fanctioDei)
ellipticas Argumeuti u exprimontur. Quod nt exempio demonstretor , Moltiplicatio-
nem e traDsformatione prima eiusqae supplementaria componaroas. Qnem in finem re-
Tocetur formola:
quam etiam hunc in modum rcpraesenlare licet;
<-,r^..„(z.,.)=/:^n....(.+-!=i^),
desiguante m nnmeros 0, +1, +«,.-., + "-y" " ^" **"* foroiula loco n pooamus
u_j ^ onde — abtt m -^H — ='5r-t .-' prodit
Digitized b,Goo';5le
$6
isti coDTeuiant valores, facto proilDCto olttinemus:
ut)i in altero producto nninero m', id allero utriqne m, m' ralores 0, +t, ±2, . .,
+ ~-~ triLuendi sunt.
At TidimuH §° praecedente, esse:
<|uani ita quocfue repraeseatare licet fomnlam:
unde iatn:
1) ■i..».. = (-l)^'y^t—'n-i«-("+ '""+'■»•'''' j.
Eodem moAo inreninit': ' '
// 1 \»»-' _ . / , tmK+tn/iK' \
s) ii™.« =y (yj ni'»("+ — ^ )■
Quae facile etiam in banc fomiam reffiguator formulae:
(,_ ==L^'li: \
V ^,^ fn.K + I.„'iK- ;
..„.n
It— k'«ii'»ni- — '--■■■»'— bI
^mK-t-«m'iK,'
TT
I, Google
ea
TT
...k.»-.«. '»■'+'■»•"'■.....„.
6) A«mnu = a«inn I 1 -
1 k-™..™ '"■■'+.«-iK' ,.,.^^
Quilms addere placet aequeutes:
't) Oiiu'ET »"K+»m'iK'
(-l)"^"
^^pi
un-t
., n»-.» «"■'+'"■'■'■
= (^) ■
9, n'^-.. '"■'+•■"■'■'■
DM-l
]u sex formulis postremis uumero m ralores tautmn positiri 0, 1, 1, S, •-., °~
cODTeniiiDt , ita tamen ut quoties m = et ipsi m' valores tantnin positiri 1 , 2 , $ , . . . ,
~- tribuantar. Et has et alias pro MoltipIicatiODe formwas iam pricu CI. ^bel mutatis
mutandis proposnit, uode nobis breriores esse licuit.
DE AEpUATIONUM MODULARIUM AFFECTIBUS.
29.
Quia eodem mode X a k atque k a X, nee non \' a k', k' a X' pendet; patet, ubi
secundum eandem legem Modolorum scalas coudas, qui in se iovicero transfbnnari pos-
snnt, alteram Modulum k, alteram G)mplementum etns k' continentem, in iia. terminos
fore eodran ordine se excipientes;
K, k. \, ...
.... X', I', \'.
Id quod in transfomtatioDibus secundi et tertii ordinis iam prius a CI. Legendre observa-
tom et fecto calculo confirmatum est. Similia cum de omnibus Modalis transformatis et
imaginariis raleant, patet, designaote \ Modulum transfbrmatam quemlil>et, aequatio-
nes algdiraicas inter k et X, seu inter u = V^k et v =: y>., quas Jtqutaionea Modulareu
nuQcapaTimns , immutatas manere,
, Google
67
1) uiii L et X inter ae commutehtnr,
Z) ubi It! loco k, X'.loco X ponatur.
Alterum iara supra ia aequationibus Modulariba$ , quae ad traosformatioiies tei'tiiet (juioti
ordinui pertinent: ■
t) u» — T* + t a r (I — u' v") =
2) tf» _ y» + 5 u» v' (u' — .'J + 4 u V {1 — «• »«) =
n}>8ervaTimus ; etusque obserratioQui ojw expruwiioues algeLiaicas pro trausformalioiiibus
supplemeotariis exhiLuimns. Ut alterum quocjne liia exemplis probetur, aequatioues Ulas
in alias transformemus inter kk = u* et XX = v*, quod non slue calculo prolixo fit. puo
suMucta obtinentur aequationes:
1> (k' — X')* *= ISS k' J." (I _ k") (1 — *.')<« — t*— V +» It' k')
S) (k* — X*)* = SUk'X' £1 — k^)a— X'){L— L'k'+L"k«— L-'k"!.
itiquidem in secunda ponltur:
L = Its ~ IM X" + 78 X* — 7 X»
^' s=
IM + tM X» -
-«SX«-
78 X*
L'' =
78 + tiS X* -
~tS*K*~^
l«X'
1"=
7- T8X"-
_ 19f X» —
USX*.
Quae^ io formam mnlto conimodiorem abeont aeqnatiooes , intrDducfeU qnantitatibntf
q ==: 1 s k% 1 = 1 — 2 X*. Quo facto aequationes propoaitae VTadnnt :
1) {q-J)«= 64(l_qq)a-II){8+<|l}
8D (q-l)- » «56Cl-qq)Cl-ll){«ql{9-qI)* + 9{45-qI)(q-J/j
= f56(l_qq)Cl-H)(405(qq + Il)+486qI_»ql(qq+il)— «Dqqli + I8q'I'|.
Quae aequationes, ubik'Iocok, x'loco X ponilur, nude q in — q, I in — I abit, immu-
tatae maDent; id quod deraonstrandam erat.
CoroUarium. (^hiia Aequationes Modulares inter q=l— ak'etl=i — ax*'
propositas formam satin commodam iuduere vidimus, intere^e potest, et i^sas fiinctiouev
K, K' secundum quantitatem q evolTcre. Quod non ineleganter fit per series:
/ q' 5.5.q* 5.5.9.9.q* \
' /q a.S.q' S.a.7.7.,- I.S.7.7.U.U.,' 1
~ "ij" ^ f "'" «.♦.« "*" 8.4.6.8.10 "*" J. 4.6. 8. 10. 12. 14 ^■••f
1 t
db, Google
K _ J |1 + ^-^ + _j__ + _______ + . . . .^
_«_ / q S.S.q' S.5.7.7.q* g.8.7.7.11 . ll.q' \
■•" JJ l.t "*" «.*.6 ■*" «. 4.6. 8. 10 "•" X.4.6.8.10.12.K "*" }
uLi Lrevjiatis caasa poaltam est
30.
FaciJiori uegolio pro traQsforniatione tertii oitliDiS ae()uationflm:
ita transformare licet, at comlatio ilia ioter Modnlos et Complementa eluceat. Obline'
raus eniui ex ilia :
(i_B»)(i + Y') = 1 — u»T«+ Suva ^u'T^ = a— •■**■)(* + <■»)*.
(1 + »•) (i _ ,•) = l_U«T»-tu»tl-«'v^«(l-ll'T')(l_By)'.
(jMil/us ia «e dactis aequalionibiu prodit:
Cl-B'){I-*')-(l-u'v')«.
1 — ■■* = k' k', * u'-
.1 — T*= X'A'=/«
exti'actis
radioibus jil:
u'"»'' = 1 — u't".
(|Ui)in ipsam elegantissimam fbrmulani lam CI. Legemlre exhilmil. Neqne ineleganter ilia
per formulas uoslras analj'ticas pn>l>atur. Qnippe e (|uil>iu casu osiS fluit:
Jl ^ k' lin* coam 4 w ( X' ^ -
db, Google
node:
^^-■zn^. ""."■ .',::::...-■. ' '■ '
unde emu »it :
k' k' + k k M^ am 4 u = 1 — k k (ia* mi li ='&*»mim.
olitiaeitfDs, qaod demooatnmdiuii'erat:
■ v^kr+ /Vat-. I.
Ut exempio aecuudo simpliciorem inter a, r, a', r «r«am aequMioiiem, ita ago.
Aequalionem propositam:
u»_v» + J «'»'£■•' -pT^**.* V (I -Til*.') «0
exhiheo, nt aequitur:
(tt» — 0(B» + 6a'T' + T*) + 4i.T(l — u't'jsbO,
(juam facile patet indnere jtosse fbrioas duas so^neiUaK;
(!.• _ O (» - »)* = - 4 U T + U«) {I - T*) .
(|Dil)iu iu se dnctis aequatioiubus prodit:
(u' — »•;• = 16u'l*(l — »»)<t — »'>=««•-•>•'» »*.
Quia &imul, ut supra probatam est, u* iu n"^ t* in v" abit,' olftiuNBUs etiam:
(V> _ a-y = 16 u" t" (1 _ u'») (1 — »'•) = 16 n" »^ u» »'.
Uinc facta divisione et exiractis radicibns, ' «iruitor:
sive ' . ' ' ' ' ■ ' .■
y'kr(v>_>o.)^«.yvr{v7.'-yTj')._ ■ i_
,y Google
31.
Alia adhuc aequaUonnm Modularium
tt» — T« + t B » (1 — n" T*) *=
«• _ V« + S tt* T* (U' — T^ + 4 B . (1 — B« %*) =
insignis proprietas vel ipso iotuita.inrefiitpr, viz. immutatas eaa^maii^re, siquidem loco
«, V ponatur— , — . QuoA ut geueraliter de ae(jii«tiq|kil>n»M(>4Hl4rilHia dcmoostretur,
adnotentuT aequeutia , quae ad alias etiam qnaestioD^^-mqi e^se possunt.
Ubi ponitur y=i.kx». obUnetnr.:,
Jy kdi ,
unde cum simul x =i , y = :
Hinc posito
nude X = sin aB|(u, k), y = siQamlku, -r-J. Hinc provenit aeqaalio:
nosm Iku. —l = fc«(o~*in'((i, t), tinde'etiaiA '
«..,n.(kB. -i.)-iMn(u. k)
& tm (kit. -i-\ s CM *m (u. k)
tf am (ku, —j = ^ cMCMm (u, k)
Digitized by Google
«ac«»m (ku, -j-| = — : ^r-
y k / im coan (u , ■)
coscoam Ik a, -r~l ^ ■ ^' f g am (u > 1)
' \.co«.(k.:±)^- -'■ '
\ ^ / ,to» cwiin j(u, k) ,
Forro peaendo i u loco a, (jttia Gomplemeiitum Moduli -r- ^' ~t~> obtiaemus adiumeato
tbnuDlaniin §'19:
lil. >m ^k II , Ji-j '= CO, co.m (u , V)
CM«in Ik*,'— — -J = tlneonD (n, k')
tg am Ikn, -r— I == uig eoam (u, V)
ctuCMin |ka, -^-} ^ (in UQ (n , k'}
^ couD (ku, —J = totgan (u, k'j.
lam iuvestigemus, tiuaenam evadant K, K' »eu arg. am {—, k), arg. am ^-^, k'j, m-
<|uidetn loco k ponitur — ; ^seu iareatigenuis valorem expressiouum arg. am l-r-, ~l~)>
arg. am 1-^, ~k~)' ^^^^ expresaiones e uotatioDe a CI. Legendreadhibitafwent F*l-pl
F ("-pi • Fit antem primum :
'^' ■ '^"f/i'-'k^-^) °//rfFl) "/TFte
Di Google
Posiloj = kx, fit
Ut alternm eruatnr intej
le - P *^ ponamtu y z=/l— k'k'x%
^> = — — - ^^' - . lam quia x iwie a. o Dsqae ad 1 omcit, »i-
/('-'•)(&-') /(■-)(■-"■■■)
mnl atqne y inde a 1 asqne k decrescit, obtinemna:
-//R)('-^) " -//('-'■)(&-') <//(--X'--)
sive nl>i k in -r- mutatar, abit K in k iK -4- iK'S.
Foiiito aecundo loco y- = cos(J), fit:
sea «bi k in -r- mutator, abit K' in k K'.
Geaeraliter igitnr mutalo k in -J- abit m K -*- i m' K' in k[^lK-^-(m-^-nl')iKj,
„.d. «a CO... j^t-i;^, l| ta .in cao, jM^ll^l^!^, ±j , id ^aod a for-
mola sincoam [k u, •y)= rincoimCu. k) ^*"
I tp(i»K+( m+..')iK-) I I
uA coam { * T — ^ — ■ "T~i..™ ~
Digitized b, Google
'75
lam igitar, posito ■ = <«, ^^^ — ■ = «»,, expressio
X = k' jiin coam I u lio coitn 4(0 tm cAarh « w .': . . ilA coam'til — tj (uj'^>*''"
mutato k in -r* io banc abit: , i .,
k" jiin coim t ai^ lin co*m 4 m^ iId coani 6 b>^ . . . *in co*in (n — 1) oil (^
ul)i I* et iusa est radix aBquatiotm Modularis, seu e Motfdtoruin uutiiero, iu 4juos pvr
traDaformatioatsm u^ ordinis Modolum 'propoailum- k ti<aDt<|fbrnlap^Jicrft.': NaoM^Q « val»-
ritius, (J u6s w iniluere potest, ut prod^at Modulus traiiKfqrmatiis, ertt etiam iBe t» . tJnde
iani causa patet, cur geueraliter Aie^uetioDes Modutaref ui^UAto . k in.-r-, X in — immu-
tatae maiiere debeant-
Adnolabo adhuc, ulu secuudum eaadem transformationui. legem qaampiam simut
traDsformatur k in k*"', X in X*"', . tjiioties't.*"* 16co fc pooatur", eUdinilD X"** aLire;
uiide aequatlones Modulares ubi »imul k in k'™', X in X^"' matatur, iqjj^ufattif-.nianffv.dje-
lent. Ila ex. g. aeqnalk) ifkX ■4- /"k' X' =: 1 , quae est ipro' tr^psforiiiMiqne ^rtfi ,ordi-
nis immutata manere debet, ubilocok, X resp. ponitor -t-^t,- , ■ "* . , uade loco k', X'
'"'■* , , :T^ .1 .i s
|io&etur ., J , , id tjund per transrormationem secundi ordiaia fieri notum est.
Quippe aequatio iTk X -|- vk'X'=: \ in hone abit": •' '
/i
Qua in ae i|)sa ducta prodit : !■-...;;.
4/"!^^= «(l + k'A') — 8kX. »iVe t>. = I+k'i' — l/'tF;
(|uae extractis radicibus in propositani redit:
AT = 1 - /k-r^ «;, /kT + /kT"= 1 . '.
Quod exemplum iam a CI. Legendre propa'situm est. Geik^raliter autem de compositione
transformationum probari potest, transfbrmationibiH duabuA aut pluribus successive adtu-^
bitb, ad eandem perveniri , (juocanque illae adhibeautur ordiiie- ' . .
K
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32.
At iuter afiieclus Aequalionum Modulariam id maxime n)«nioral>xle ac siogulare milii
videor aiiiniadvertere , quod eidem ontnet jiequationi Differentiali Tertii Ordinia tatufaciaat.
Cuius tamen ioTesligatio paullo lougius re|)etenda erit.
Satis DOtuDi est *) , poaito a K -i- l> K' = Q , fore:
dwiguaatiLus a, b Constautes quaslibet. Ilacliainposito a'K-i-b'K'=Q', desiguantiljus
a', b' alias Constaules quaslibet, erit
, {q J!21 _ « j!12.> ^ „_ .1., Jn J«: _ n- Jl«_l
Qailius combinatij aequatiouibus, obtioetur:
aode- iulegratioHe fiicla:
Conslans C a CL Legendre e casu speciali inventa est = — -^t unde ianj
— i-,(.b--.-k)
•<3'
V
Kl-l")
— L,(.b--.'k)jk
Q Kl-k-jQQ
Similiter designanttt X aliom Modalnm quemlibet , erit posito a A -4- |8 A' = L ,
«'Ah-i8'a'= L',
-4— (-«'-''«''*
L ~ X{1— X*)LL
Sit \ Modulus in quem k per transformationem primam n*^ onliuiii trauHfoniiatur ; sit
porroQ = K, 0' = K', L = A, L' = A'i erit;
L' A' _ nK' nQ'
~T~ ^ A ~ K ™ Q •
•) Cf. Ugmdn TniU dc* F. E. Tow. I. Cap. XIII.
Digitized b,Goo';5le
node
Ddk _ dx
k(l — fc')KK ~ A,(l — V,AA ■
Inrenimas autem pro ea transformatioDe A :=: — jrp , uude i
MM = —
xci— x^Jk
• k(i-k')dn '
Ak
ndX,
k{l_k*)KK
\(1— X.TA^A,
onde el hie:
MM^ ». V>-\')" .
Gcnerallter aatero, qaicanque sit Modalos X, sive realis slve imagiiiarius, in quern per
transfonnationein u*' onlinis transforDiari potest Modular propositus It , ralehit aeqaatio :
MM- * Wi-jOii.
"" = T- x(i-x-)dk -
Quod ot probetnr , adnotabo geoeraliter obtioeri aequationes formae :
desigoautiLas a, a', a, m Dumeros impares^ b, b', j8, ff numeros pares, utrosqne positivos
rel negatiros eiiumodi , nt sit a a' + L b' = 1 , »»' -i- &ff ^ i *). Hinc posito ;
, • K + ■ b K' «= Q , •' K' -)- i b' K s= Q*
«A + i/lA' = L. «'A' + ifl'A = L',
olilinemus, quia aa' + bb' = 1, ««'+|9|3'=l:
Q' — nifdk L' — irdX
Q ~ JkCl— k-JQQ ■ "TT " JX{I— X'JLL '
*) Accuratior numeramm ■, ■', h, h' eel. eel. delarminatio pro «lnguli« ciuidem ordinis moirormttiooibui pa-
vibui labonra diflicullslibni yidetur. Immo baec detcrioiiuilio, niii cgregic faliimur, niaxinic a Itnitrbui pen-
del, inter <|u«* Modulu* k Tenalur, ita ut pro limilibut divanii plane a&a evadat. Id ^nod quam inlricaUm
reildat quaestiooeni, eipertaa cognoicet. AnI* oninb BBtein accurattai in Mturam Modulorum imagiaarivrum
iiii{uireDdum ene otdetur. <inae adkuc Iota iacet qtuutio.
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»
Aduotabo adhaC) aeqaationem inrentam ita quoqae exhil>eri posse:
HH =
X'(l— X')d(k') _ 1 .A.''a^A'')J(kT
k'ci-k')dM n : k-(i_k")d(>.") ■
uude videtuus, expressionem MM Aon mutari^'ubi loco k, X Compleiueuta ponuiilur k', >.',
sive quod supra demoDStraTimas, transfiarinatioiiiliiu complementariis , sigoi ratiooe non
haliila, eundem esse multiplicatorem M. Forro mutaQdo k in X, Xiok, quo facto trans-
formatio iu sapplementariam abit, mutaturMMin ^
n ' \C1 — V>dk "" ddHM ' oM '
quod ef ipsuni.pMpra probatum est
33- : .,
Posito Q ^ aK H- 1)K.', L = «A + i^A't Go«i»tantes a, b, «, J3 ita semper deter-
minare licet, utsitL=-^, sireQssML. Forro habentur aequationes:
I) (fc-k')-^ + a-8k')-i^_kQ =
quas eliam huoc iu modum repraeseatare.lic^t:. , - ■ . .
(k-k')dQ ■ ■ ,• . : .- .1 . .
dk
dk
<I -
dX
Substituamus in a/equAtioiie;
»-'')-if + ''-""'.4r-'."'-
,, Google
Q = ML, pradit:
qua per M mnltiplicata, obtinemus:
5) LMfd-k-j^+a-Jk')
dM
dt'-
-kMJ + d -
dk
" dk
At
e $° antecedente fit;
D(k— k')dX '
uude
{l-k')M"dL
dk
= Jt:
-X')dL
Po
rro ex aequatione 4) fit;
'{^^^^}
= XLdX. unde
(k— k')M'dL
Ji:
-X>)dL
d>.
XLdX
ndk ■" ndk
Hibc aequado S) divisa per L in banc abit: " '
«) « {c-'") 4^ + <■-»•> -^ - '«} + 4lf = »■
Ubi io hac aetjaatione valor ipsias M ex aetjuatione M' = -■ ~ .,. . - substituitur, ol>-
tinetur aequatiio dif&reotialis inter ipsos Moduku k, X, qaatu facile patet ad ordinem ter-
ttom asceodere. Facto caJcuIo paollo molesto iaTeoitar:
Sd'X' tiK dU . dA.' //1+k'k' /1+X'\' d*.')
^ -dF--"-dr-7rt7FUfc=^)'"tnn?)--dF; = ''-
In hac aequatione dk ut differentiale coD^tans coosideratum est. Qaam ubi iu aUam tmu-
fbriaare placet, in qua di£Ferentia]e nnllam coostana positum est, pon^ndam erit:
d'X d>X dXd'k
IF ™ dk' " dk'
dU d*X fld'Xd'k dXd'k , SdXd'k'
onde:
SdX'd'k' . JdX'd'k
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Hitic oequatio 7) maltiplicata per <Ik' in sequeutem abit, in qna (Ufiferattiale nuUam
con.slans.posLtQm est, vel in qua ut tale, qaodcunqae placet > consiilerari potest:
Hanc patet , elemeotis k et X inter se commatatiit , immntatam mauere ae<]uatioDem , id
uuod supra de Aequationibus Modalaribiis probavimas.
Operae pretium est, alia adhuc mpthodo aequalionem illam iTiffereultalem terlii
ordiuls in-^eiitigare. Quern io fiaem introducanius in ae(|uatioDem, uude pro5ciscimur:
d*0 dO
(k-k-)-J- + (i-»kO-j^-i«.o
quautitatem (k — It*) Q() z=s. Fit
-il. = (I-SklQQ + «(l-k') Q i5;
-i^ = -61QQ +4(I-Bk')QaQ + «(k_l,")(i^y+ «(l-k')g-^.
Oua in ae<|aatioQe uhi pouitur;
(t-k'l 4f- = kQ - (1 -« k1 12- , prodit
^ = -"«« + «<'-»■)« 4f + '<'-"'(4r)"
•= S-12.{(l_Sk-) Q + (k_k.)12.} - «kQQ.
Qua aequatione dacia in 2s = 2{k — k*)QQ, oblinetnr:
il£l = J(k-k')QH.|j(i_jk-)(}Q + t(k_k')Ql2.|_«k'(4-k')g'.
sire cam sit:
• (k-k-) Q 4f = -JT - ""■"'"'''
J(l-Sk1 QQ + J (k-k>) Q -12. « -ll- + (l-Sk-jgQ.
obtiiiemua :
Digitized b, Google
lam vero posito a'K -f-b'K' = Q', ~- ^ t, Tidimiu eftse -jr- = -pr — |^.p- = — ^»
desigaaote m Coustaotem; iuul« s ^— — . Aequationem 9) in aliam trausfonuemus,
ia qua dt coMtans positnni esL Erxt -~ = -J~-, ~ = —^ — ^^ ; yuibiu
Hubistilutu ex aequatiooe 9) prodit:
■dI^d7-'d?al7+(krri7?-=*** =*»*'«
10) td'kd k — »a'k* + /lt^j'dk* ssOi
ubi secundain t, quod ex aefjoatione evasit, differenlianduni est.
Fouendo — . 1^ . r-t^ ««, Coostaotes a, 0, a, ff, quoties X est Modulus traua- *
fifftuatus, ita determinari poteruat, ut sit t=w; nee non umili modo oLtiuemns:
11) td'XdX — Sd'x< + ^i±^Jd>.« = o.
in qua aeqnaliooe et ipsa secundam h := t differentiandum exit. Multiplicetur aetjuatio
10) per dX't aeqnatio ll) per dk*: snbtractioue facia obtioetur:
iX) tdkdA.|dxd'k — dkd'xj — sjdx'd'k* — dk'd'A»l + dk'dA*|/^^^j'ak' — (^l^l'dx*! = o.
At haec aecjualio cam aequaliooe 6) conTeuit, iu qua scimos, differeotiale qaodcunque
placeat tamqaam constans considerari posse, ideoque etsi iuTeula sit sappositione facta,
dtesse difiereotiaJe constana, ralebit etiam, quodcunque aliud nt tale consideratur.
Ecce igitur aequatiouem differeDtialem tertii ordinis, quae inuumeras faabet solo-
tiooes atgebraicas, particulares tameo, viz. Ae<iuationes quas diximus Modulares. Atlit-
tegrale completnm a fuDCtionibua ellipticis pendet ; quippe quod est t = i* , sire ■ * ^^ . g r-
:= — _|_ ■ .T , quam ita etiam repraeseotare licet aequationem :
mKA + m'K'A' + i&"KA' + m'"K'A » 0,
designantibus m , m', m", m'" Conatantes Arbitrarias. Quam integrationem altisslmae in-
daginis esse censemus.
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Inquirere posaemus, an Aeqaationes Modnlares pro traasformatioDibus tertii el
tjuiali ordiais reapse, quod debeot, aeqaatioiii Qostrae difierealiali tertii ordinia satis-
facisut. Qnod vero cum Dimbi prolixoa calculos slbi poscere rideatur , idem de tramfor-
matiooe secaodi ordiois, nbi X = ~ , , demonstrare snfficiat.
A.
kk + k'k'=c 1
-s
dk — k'
dk' ~ k
dk'"
4
(I+k)'
d'k —I
dk'- = k
k'k'
k»
d';i
_«
d<k 3k'
-dTT
(i-t-Vf
dk'. k. ■
t;
ak'd'X'— dX'd'k' 16
AV k'(l
k'k' 4
+I.V k-(l+k',>
4jk<k--(l+ir)
4[k'-(l-k')'-l}
k'<l + k-)-
k'(l+k')'
Porro obtinetar;
dkd'X— axd'k Kk*
. 6k' 6k
(s(i-kV+il
dk'< ' k(l + k')'
■• vd+iy
k.(i_ky
unde
Potio iit
4ikdA.{dkd'x— axd^k} «k*'(«(i— k7— i}
Ik'- = k"Ci + t'j* '
3[dk'd'X'— dX'd'k'J — tdkdx{«lkd>X— d^d'kj «{«k'' — 1)
d k'« ~ k'(l-|.k')4
\k— kV "dP^ k«k''
( i + k' l' dk' _ ft + A-'l' dA.' _ B(l— tk-)
U-v/ dk" U— >.'/ dx" ■" k'k"
dk'dX' (/ 1 + k' i' dk' ^ / 1 + X' v dX' ^ _ l>(l-»k'')
dk*' Uk— ^V dk''~lx— X',' ~ -
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Hinc tandem fit, quod debet:
dk'«
*•" dk'« (^k— k»; d^ U-x'/ dk"/*^ k»(i+k')» + kvi+kO* "
Ubi methodi expeditae in promtn esseot, si qnas aeqnatio diflferentialis solationes alge-
braicaa habet, eas eraendi omQes: e sola aequatione differential! a noLia proposita Aeqna-
tiooes Modalares, quae singulos transformationam ordines spectant, elicere possemm omnes.
Qaam taroen materiem ardnam qui attigerit, praeter CI. Condonet, scio neminem, atten-
tione Aoaljittanim digoam.
Aequatio sapra inrenta:
MH = — .
34
X(l— XX) dk
k(l-kk) ' dX '
cuius ope ex Aeqaatione Modolari inrenta statim etiam quantitatem M determiaare licet,
digna esse Tidetnr, cai adhnc panliaper immoremor. Non patet prime aspectu , quomodo
valores quantitatis M in transfbrmationibns tertii et quioti ordinis inventi cam aequatione
ilia coQveoiant. Quod igitnr accuratioa examinemns.
a) In transformatione t«r<Hordinis, positon=vk, r^/ji inrenimns:
1) u»— »« + tii*(l— u't') = 0.
qnam ita qaoqne exhibnimns aequationem $. 16: ■
Porro fieri Tidimas:
Oifferentiata aequatione l) obtinemus:
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an
sive loco 3 posilo I " 1 1— — ^J:
*)
g^'-t l+u'v'+gu'%
Su'+» l + o'v' — 80T'
Ex ae(|U8tione 1) a«<juitiir:
1 _ u* = {1 + o*) [l - ,' + *UT (1 - u'y*)}
= I — u*Y» + u« — v'+lu»CI + u*)(l — u'v'
= l-u«T*+2u>v(l-u».*j = (l-u'OCl+u'v'+Su'v).
E^dem mode iDrenitur:
nnde
~' = ■ .* .T-"^ ■ »"e ex ieuaatione 4) :
1— u' 1+u't'+»u>v ^ '
t — y' du _ a^' — u
' 1— u* ■ Av ~ ta>+T '
Qua aet|natioiie dacta in
3U - (t«' + T){tT'-l.) '
prodit:
S * u(l — n') 'At S ' kCI — kk) *^V»+*"'/
Q. D. E.
b) In transfomiatioae jui'/itt ordiDis, positou=r k, t^vx, iuYenimtis:
1) u»— t« + 5ii't*Cu*— v') + 4nvCI-u«v*)= 0,
quam his etiam modu exhibuimus aequatlonem $$. 16. 50:
*' ud+u'v) TCl-u,')
S) (u'— 77- = 16u»T'(l_n')fl_v').
Porro inTenimns :
»_ut 5u{!+u'v)
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Differentiata aequatione S), obrinemus:
6uv(l— u')(! — »')(iiau— »dv) =
„(u»_,'){l_B')(l-5T')dT + »Cu'-»')(l-y')Cl-Su')du,
are:
AequatioDo 1) dacta in n% t% ernitur:
5u'— u" + T*— 6«»T« = (1 — i.«v*)(v'H-5tt* + 4u'T)
5t*_v» + u'-Su' »•=.{!-«' »•)("' + 5»'—4u»*).
uode aequatio 5) in banc abit :
v(l— *') Ju h' + 5t' — iuT*
•^ u(l_u*) ' dy = ^ + 5u' + 4u'T •
Fonatur u-»-v*=3A, n-»-u'v=zB, v — u* = C, v — ut' = D, iuut:
-^ = 5, sive AC = 5BD
C 5B
„' + S,'_4u.» = uA + 5yD
t(I— v'l du dA + 5tD _ uAB + tAC _D_
^ u(l — u') ■ "57" " »C + 5uB ~ vCD + uAC ' B
bB + tC ad _ AD
~ vU + uA ■ BC BC
Fit enim:
uB + yC = tO + uA = uu + TY
Unde etiam:
= -^ = 5MM.
MM :
1 v(l — v') du _ 1 K{l — \K) _dk_
~ ■ u(l — u') ' dv ~ S ' k(l— ktj ■ dX
O. D. E.
L 2
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S4
THEORIA EVOLUTIONIS FUNCTIONUM ELLIPTICARUM.
DE EVOLUTIONE FUNCTIONUM ELLIPTICARUM IN PRODUCTA
INFINITA.
35.
Proposito Modulo k reali, imitate mlnore, videmiu Modulum
K = k"Ln GOim — tin eoam — . tin coin f''-'>^ t*
I n n n f'
in qoem ille per transfonuatiooem primam n** ordiBis mutator, crescente Domero d^ ce-
lerrime ad nihilum coDrergere ; adeoqae pro limite n = 00 , fieri X =3 0. Tnin erit
A=-7-f am(a, X)=u, unde e formuUs A = — ^, a'=-=7^ obtiaemns:
M — ^ —=— — —
FoDanius iam in fbrmolia pro transfonnatioue primae snpplementaria §. 26 ^ loco n,
n=00: abitam(-^, x)inam(^, ^) = -S'' y = »inam(^, x)insiii^i porro
am (n a) in am (u) . Hioc e formuUs illis nanciscimar seqnentes .*
8Kj \ ■°-ir l\ ■'°— ir l\ "°-ir /
V "°TirA ""TirA "°-iK-/
('—fi^]i'—fiT^)(^—fkA ■ ■ ■
\ ""-nr/V •'°Tk-/\ "-jiT/
Digitized by LjOOQ IC
V ""-jiT/l "• TS- /\ "'-sir/
A am u = : :
V ""-JK-Zv ""-sr/V "-tr)
V ""-iir/v ""■-«- /\ "'-jKr/
/ iitK' SiwK' Siirlt' \
— nn? . . »i.K' — + — sTiTT I-
""■-SK~" ■"•-jir-" ""Tk— " /
— wK'
Fouamiu io seuuentibns e '^ = (f i -j^ = x , aive u =5: , node
* : sin X ; fit :
ande :
1— «q*'"cMg« + q«°
db,Goo';5le
SB
«,'»J»I'
l+!,-»eo.«. + ,—
(!+,-;• =
(! + ,■-)•
!,».io.
1±S,".».+,-
■L .+,.» -
1+q'"
!q"(l+q"»)
1 — 2q""M»*x + s:''
«K' 1— Sq"»cos2i + q'"
His praeparatis, atque posito Jjrevitatis cauiMi ;
A =
l (l-q)a-q')(l-q').. j"
l(t-,1(l-qV(i-q*).. /
_ 1 (i-q)('-q')(i-T)-- j'
I (i+q')(i+q*)(i+q')-- I
^ ( (!-,) (1_,1C1-,-).. )•
( (i+q) (i+q') (l+q'l ■ I '
prodeuot FuuctioDnm Ellipticaram evotatlones in Froducta Infiaita fuudamentales :
«AK . (1— aq'coi«. + q')(l-8q'l».»»+q'|(l — »q'Coiti+q") ■ ■
1)
2) coiam -
S)
(l_!qco.!«+q')a— «q'c».!.+q')(l— Sq'<».J<+q»)
(l+aq'eo.a. + 1-)(l-Hq'co.«. + q-)(l+«q'c«.8,+q'')
' (t— 2qco(«»+q'J(l— «q'coi2x+q'HI— «q»c««i+q")
(l + 8q=..8-)-q')(l + »q'eo.i.+,-)(l + «q-pi.t. + ,")
— : — skt
/TIT- *''"
1 — k Sin Bin
(I— 8,cii.!.+q')(l-2q"coi!i+q'')(l— «q'cM!.+q") .
+q')(l— 8q'nn.+q')(l— «q'»iii.+q') .
-H-q") (l+8q'».. + ,')(l+aq'«~,+q>) .
Jt (l-8/"qiilli + q)(l — 8/V.iDX.fq')[l— 8/V,lD.+,').
(i+a/T<i»'+q) (i+8/V"««+q') (i+8/V"»«+q') ■
, Google
Nee noQ aliad formulanim systema, qaod resolationem propositamm in fractiones aim-
plices suppeditat:
t) •"■ '■ .in-f ■^('+.'^ I /V('+1-) I /Vd-I-T) \
' . kK ^ 1— tr,coi>> + <l' 1— «q'co.«. + .|' ^ l_2q'co.«» + q'' ^ 7
n .uiiii '"' '" mill '^''— C' /V('-1') I ^1.1— a \
' IT kK \ 1— Xqco*Zx+q* i— S q>co«Xi+()* "*" 1— t<i'c(.»8i + q'« V'
Qiii}jus adilimDs «x eodem foote maiuiDtes:
, .„!Ji;»±. + .A,.^.ii^l!);si._.A„,g.ii±321Si + .A,e^,»±a:)i£i_..
If 1 — q 1 — q' 1 — q*
Id formula postrema sigDom soperins eligendam est, (jaolies io termino uegativo, inferius
ijiioties ID termino positiro computationem sistis.
36.
CoDtemplemur formulas l) — S), in ijnibusaate omDia (juanlitalutii , quaa [ier A,
B, C deugnarimas valores eruendi sunt Facile qnidem iuveuitur poiiendo x =:= -^ , c
formnlis 5) , 1) :
k- » cf f-l)('-i')('-i')--- l'_ cc
la+«(l-q')(i+q')-.. (
mde C = /V 1
« Ul+q)(l+q'
r)('+q') ■ ■ ■ 1'
') (l+q'J • • • /
uade B = . At ut ipsius A eruator Talor, ad alia artiiicia coufugiendum est.
Ponamnse";= U: uhi x iu x+'.|^ mutatnr, abitUiu/^Uj siuam — -^in
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88
E formula -1) antem oblmemns:
__«K. _ \K I V-V-' \ {a-^/VHi-l'V^ . .}{(l-q-U-)(l-q'«-) . .)
" ' ' \ t ) ia-,u>)(i-,'Di..}la-iu-)a-»'u-)..j •
unile mntando x in x H — jjt- :
1 ^ AK / /^U-Z^U-h {(l-q'U-Xl— i'U^..|{(l-<|U-)(l-q'U-')..|
k,ia.m— * " ■ ' /ifl-q'U')Ci-q*1^..j{a-U-')<l-q'U-')-.)'
qnikm in se ductis aequatiooibus , cam sit:
prodit :
1 1 /AK\' . ^VT J fKA iVq
— .= -—I ). sive A = ■ 7_* i nude — = — - ^ ■■
Hinc erit B = ;; = s/q y p. lam igitnr fit:
IKi 1__ gy^'in'q— gq'cMt»+q*)Ct— itq*M'»»'+q')»— tq'coili+q")...
.mam ^ - ^. (t _8,««4« + q'){l-*q»«..l. + q')a-«q*c«li+,") . . .
2K. _ /T^ aWco.«(l + 2q'cw8x+q*)a+*q*co*»«+q')(l+»q'cwt.+q^...
coon, ^ _y ^ . (l_Sq«,.8x + q')(l-»q'c<..t< + q«)(l_,'«»t.+q»)rr:
8K» _ rp- (I+aqcMg. + q')Cl+»q*M>»». + q«)(l+»q'«>»«.+q») . ..
jr ~ * (t — «qeo.2i + q')(i— Sq'co.tx+q'ia — Jq'co.tr+q") ... *
AeqaatioDibus in »e doctis:
. (i-K-ja+l-Kl+q-l ■
~> (i+l)(l+q')(l+'l')-
prodil :
/T ^ ja+qXl+q-Xl+l")-}'
db, Google
lam rero secandum EuUrum in IntrotU (dt Partitione Kumerorum) est:
(l-V)(l-,')(l-<l-)...
(l+qXl+s'Xl+q") • • • =
(l_,)(l-V)(l-q") ••■
(l_,,)(i_,^(l-,.).
1) {(1— iJCi-q^d-q-Ki-q') • •)' = 'VL''
Advocata formala;
'Vl _ I (1— iX'-sld-l') ■
/X:k ' (1— lld-lVCi-l*) ■
tkk'K'
«) ((l--l')a-'l')(l-q')a-q') •■)' = 7=r' unde etiam:
"•V q
♦/Tk' k' K'
») {(l-q)a-<rt(l-q')(l-q")..J' = -
.•;^
Quiltas adderc licet, quae facile sequuotiir, formnJas:
«) {(i+q)(>+q')(i+q')(i+q') ■ •)' = ~7^
5) ((i-H')('+i'l(i+q')(i+q') ■ ■ }' = r4- r-
4V fc V q
6) {(i+q)(i+q')(i+q')a+q') ■ •)• = — ;ft^ ■
E tjaibns eriam colligitur:
■■ «irq
d+q-Xl+q SC+q') ■ ■ r
(i+q)U+q'r(i+q') •■ I
„ ., _ I (i-q)(i-q')(t-q')- f
' I a+q)(i+q')a+q')--(
8K _ I (i-q')li-q')(i-q'i ■■ \'\ »+q)(i+q')(i+q')- i'
" ir~ ( (i-q)(i-q'Hi-q')- 1 Mi+q')('+q")('+q') ■• 1
4 y- Hi-q')('-q')('-q")-- j'
' ' I (!-q')(l-q-Ki-q") •/
M
Digitized b, Google
g-tld-l-Xl-q')
I a+q)a+q')(l+«-
/
(<1-
-,■)(!-
-,•)(!-,•) .
1(1-
-■Od-
9')y-q').
iiz
■«1-
q'Xl-q') . .
c.
« Ui+q')(i+q')(i+q-)-
E formulia 7), 8) seqnitur aecjuatio identica satis aLstrusa:
") ((i-q)(i-q')(i-q') ■ • j" + wqlci+qia+q-JCi+q-) • • j"
{(l+1)(l+q')(l+q')-)".
37.
Vidimus supra, ubi de proprietatibua aequationum ModularioiQ actum est, diu-
tato k in —, abire K in k (K + iK'}) K' in kK.'; porro fieri:
Mm [i n. -j-j = CI. CO.I. (u. k-)
cotun |l[ D, -i—j = «n coam (u. k')
A un ^k„ ''■ \ '
'"■"('■"■ k ) A™(»,kV
Commntatis inter se k et k', hinc sequitur, ubi k' in
-TT- seuk
K in k'K. K' in k' (K'-»-iK); porro fieri:
liii am IV'u, — ^j = CO* cwm u
"""(-•4) = —
i.n./k-uiL^ '
\ ■^r) A.mu •
uode etiam:
, siDiul altire
.(-4)-i—
db, Google
91
— «K'
At mntato K ia k'K, K'in k'(K'-hiK), ahit q^e in — q, untie vice rersa fluit
THEOREMA I.
Mutato q iu — q abit :
K in V K, K' in k' (K' + iK)
simol q in — u, x in -; x, abil:
Inquiramus adhuc, qnasnam Functiones Ellipticae, mutato q vel in q' vel in
y^q, snbeant mntationes.
Vidimos supra, Modulum \, per transfonnationem realem prhnem n** ordinis a
Moflolo k derivatum, ea insigni gnudere facultate, ut sit;
n. ^ ,
unde matato k iu X, al)it q = e in q". Idem, a nobis de transformationams
DgilzedoyLjOOglC
imparis ordiuJs geueraliter probatum, inm dudum a CI. Legeudre de transforinatioue se-
condi ordiDis probaluni est, videiicet posito X == ., ' fieri:
node ridemus, iiiutalo k id ~ , abire q in q'. llinc vice versa obtineiuus
THEOREHA II.
„MuUito q in q' al»it k in -jxp*) ^ "• { — i — ) ^*'
uode
etiam:
k' in
.+1'
k'K
in ^VK
V^
'"^
<n.
,Ki.ii
1+kin
1 + k-
l-l in
«k'
1+V in
c+^y
i+k'
(i-^V
Ex iiiTersione huins theoreiiialis obtinetor allemm
THEOBEMA III.
„Mutato i| in /"if, abit k in -jSr • ^ '" (''>''') '^>
unde etiam:
-»4^
.^kiniy^
--^
■ -kinJi^
kKinZi^.K
--■T^
,^'K in k'K
1 k-i. " .
1+k
Quae tria theoremata evolutiouibus §§. S5. S6 propositis inulliuiodis confirmaiittir, .-iuam-
que in seqaeotibas frequentissimam inveniuot applicatiouem. Quippe quornni ope vet
ex aliis alias d«ivare licet formnlas, vel alinade ioventac coDimode c<mfiniMiitar.
oy Google
3&
Qoaotitates, in qttas posito q"" loco q abeunt k, k', K, designeDiiu per k*"", k""'',
K""', ita ut k"°* sit Modalos per transformatioaem realem primam d** ordinis erutas, eius-
(jne complementum k""^ FoDamua in aequatione:
I (>+qKl+l')(l+q')(l+i').../
loco q sacce&sive q'^ q*, q'^ q'^, cet. , prodit facta multiplicalione infiuita;
J~ l.'ri,.yk,.ri„,y = f (t-,')(l-q')n-q')(l-l-) ■ ■ ■ I'
"■ I <l+rt(l+'l')(t+q-)(l+q")-:- I'
at invenimiu: »
I (i-q')(t-i')a-i')(i— !■) ■■■ !■ t/Vit
\ (l+rtO+q-Kl+q-Xi+q-) ■■■I" » "
uude;
SK /k">'k«''k<')'k<">'
" — = V a •
Cum sit k"''= -f^, fit ex 1):
/»K\*_ 1 tyk' t/'k^v" i/"^*^ t/"k">'
\ , )-T' i+k' l+k'-' • i+k"!' ■ i+k"' ••••
ande divisione facta per l);
Quae etiam eo obtinetar formula, quod sit;
*
1
JK<»>
t
■l+k<«
db,Goo';5le
94
ande cum^ cresceute m in :nfimlum, limes expressionis sit 1, facto producto in-
finito prodit 2). Posito:
m = - , ti — V mn
fit:.
l+V
.A-
unde :
t
i+k'
ideoque :
sea desigoaDte ft Umitom commutiem , ad quern m"", d"" coiirergunt , crescente n iii
infinitum :
Quae abnnde nota sunt.
Ponamus rursus in foruiula :
(I — SqCo.SK-f-<l")[t— aq'co,2« + q«)(l--«q»coi«< + q«').
y Google
85
loco q successive (j^, q*, q', eel. ; sit porro :
Facto producto infinito, cam sit:
obtioeoius :
,r a— Sq*«w«x+q«)(l — iq'cwai + q^Cl — tqOcoiai + q") .
lam vero e formulis:
» yY»iDi(l-tq'cwg« + q')(l — gq*c.>«8. + q')(l-8q-CO»«. + q"J
y-k ■ (I— JqcMix + q^a— 2q'M.»Si+q"Kt — *q'e««» + q") ■■ •
'/?
W«>.»(l+aq'co,g» + q*)(l+aq«co.2» + q')(l+aq-cO.»x+q'').
(l_tqCO.*. + q')(I-«q»-Co.ai + q»)(I_q'co.», + q") . . .
1 ta..6--a-»S'«'«»- + q*)a-aq*CO»2l + q")(l~gq'C(.»8, + q") .
^k' (l + «q'cM«< + q«){l + «q»CM«. + q')fl+iq-C0.8»+q-) . .
uude prodit formula memorabilis:
CKi
S. tg »m —
4) bngx
Ut eandem per formulas notas demoustremus , adTocemus formulam pro traQalbrmatioDe
secundi ordinis, qualem CI. Gr-iua exhibuit is Commeutatione inscripta: ^^ Deter minatio
Attractionia " cet. :
db,Goo';5le
96
cjuae lirevitatis causa posito:
.„(i5r:i., i.».)=*«., ^«^{^^. k-) =
ita eshibetnr:
node eliam:
MM 0"' a"'
I + k"> .in' *<•)
1 — k'" »itt' ^"*
(l + k'")ig»'"
Formnia postrema ita quoqae repraesentari potest:
IT If
DDde loco q successiye posito q', q*, q*, . . ., quo facto k, K, <p al>eunt io k'"', k'",
k*", ..; K'", K'*\ K'", ...; q>''\ 0'*\ (t>"\ ..., obtinemus:
tit'*' «K<'> ' &'•'
«K"" ■ A"*>
lam limei expressionis
/ ex'PV
/ ex'P>i ,, A
d by Google
97
cnsotnte p w infinitmn, fit
tang 1 1
turn enim fit k"** = 0, K"" ^-j-' am (a» k) = n; lunle iem facto prodocto infinito
et posito, at supra., 5 ^ &'** &'*' a'" . . . , prodit:
HP _ *»'
({uae est fortanla demonatranfla.
E formula:
Slalb
Algorilhmus doo inelegana peti potest ad coniputnnda lategralia Elliplica priiiiae ^wciei
indefinita; idqne ope {ommlae, probata lacilis:
Quern in finem proponimns
THEOREHA.
/^
' /'minCoi9' + o n Sin ^'
/ ■ilinCM9''f-(inSin ^' = A , •
fonnentar expressiones ;
7 =" '''"■' =" '^ =V — ;;rfa —
_I .n /». ==. '^=V— S+S'—
m'+n" , r-S-Tr « ..- _ / m"nr(A- + ..-)
X = „ /„» _. ^ =V ■'■' + 1'
N
db, Google
designante ju limitem communem, ad quern quantitates m"'*, d"", n'^* cnscente p f&<
pidissime conTerguot, erit:
, a' A' ii" . . .
ungfi« = p-B ■ttitg^.
liiideiii methodis, quiltus ia aQtecedentibos usi aumas, inrenitar etiam valor
product! infiuiti
iV^i gW »VX* iVX'
/'k /k^» /"i?" /*il""' . ■ . •
Quem in fiDem allegamus formulas §. 36, 4), 6):
quarum posterior e priori oaacitar loco q posito successive q', q% q* cet. et facto pro-
dncto iufinito,' unde ohtinemus: j
k g'/V »V^ t/v
4/Tr/*q" /"k"' k""' /'k"> k'**" v^k<'> k'»»'
latn rero emimns l):
-T- = V i- •
unde :
5) ^ "^ ''^ '*^ **^ a'/V
Quae licet aliena rideri possint ah institato nostro , cum nee degantia careaot , et
magnopere faciant ad per^jiciendaoi nataram evolattonam propositamm , . opposnisse
inrat.
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«6
EVOLUTIO FUBCTIONUM FLLlnlCABUM IN SERIES
SECUNDUM SIHUS TEL COSINUS MULTlfLOBUM ARGUMERTI
rROGREOIENTES.
E fomiulu Kupni tradtlis:
tKx tW . (I — gq't
.». + q-Kl-»,'C0.«, + ,-)tl-8q'C0ll. + q") . .
■yx
■ (H*l'»"«-+q')(l+'q-f»«»+q')(i+«q'c<»»»+ q") ■
(1— «qco.JlH.q')Ci — tq'cO.«. + q«){l — «q»colf. + q") .
«K. ^/-p- (l+iqco.«i-t-q')(l+iq'c°'«'+q'»'+«q'<»'«-+q'T .
1 — Noam /"- r
'+""■"—
— r «if^
1 — k iiD am
(I-_tqeo.«.+q')(l^»q'«..«. + q^{l-tq'co.!,+q").
(l-«q.l.. + q',(l-«q-.to. + q1(l-t»q .i.-fc,-)...
(l+«,.ioi+q-)(i+sV'l««+«(i+«'l'"»-+q') • • ■
(i-»/7""+qH'-«/i'»"-+q')('-'/V'i«-+i')
tl+«/"?-in.+q)(l+»/V'i»-+«('+«/q'"«'+q')
logarithum singuloruui fadonun in altera ae(jaatioDtiin parte eroluUSj post reduclioiws
obTiaa, MquuDtur hae:
6) log fiD •m —
7) loKcwun-
S) kfiim i
9) lo« . / -
- I ''A'.l'
«q
».!.
1
+ , '
•l
».t. 1
1
— 1
4q
^.t.
1
-q' '
- +
«'""
•a+q")
«q''~«-
«('+q')
'(l-q-)
»(l+q")
»(l-q')
4q*CMlOi
100
""" , . 4/7.1.. 4/7i.». «/y.i.'5. ■ •
Oaibiu fonnalis differeotiatia , ubi aduotamu* formulas dilferraitialcs probata fadles:
d. log «««.—-— ,.,^ «».»-—
«K '^""-T- . lit h^r ]
V . «E
enumas seqneDtes:
4q.int« 4<|'dD4.
1+q ~ l+s"
4q.i°«. , 4,- .1.4.
IS) -?i^ ■-.■■^■■: "■_ 'T"*' i gq-ne. II,.J.I0
-+■•
Google
tS.% <».«^ 1 — q 1— ,» ^ 1 — q'
ISI *^* An— '"^' --r */^«"' «/iFw8. 4/yen.S,'
* ' « 1-q . i-q' ■** l-q*
UIri in hnfoftnulis loco X ponltur -| x, ermtnr:
» 4q«ImI. 4q'«p*i 4q'«n6i
^ tK> - *' i + q + l + q- l + q>
4q«iiiSs 4q*«ui4i 4q'nii6i
= coi>« H — a 2-— J 3 J—
1 — q l + q" 1 — q*
4qHai 4q*uiiSi 4q*da5s
Formula 18) poDendo -j^ x loco x immnlata maoet.
Mutaudo q in — (| e thefffemnte I. $. 87 formulae tl), 12) ia 17), IS) ab-
euut; 18) immntata maoet; e formulis l4)j lA), 18), 19) obtinemns:
IKk 4Vqcoi» 4/^q'eo«8i 4^^coi5x
* * l + q **" l+q' **" !+;?
4q'iiii8ii 4q*«ipSK
tKi «/^«in i/^ting* 4V^HJ|5»
'~^~ l+q ~ 1+1' . f+?
,, Google
tOf
- Formulae 19), Zl')':p^ evQlatioMS ttp1as_ ex_'.'i^^ J^t^^V^ fticUe derlvan possnnt,
(juas supra attulimus $. S6. 6), 7): - ■ .
«K. _ _«^„ / /^ft-q) /Vo-1') . /7a -q') _ \
E formula 9) §-35:
± - + «-..((4^)'-)- »-«(({^j'-) + '— ((j^j-) - ■ ■
setjuitur adhuc: ' ' " .. ' — "*■ ''■"'" '".". " ' ^-
Eandem eoim pro signi ambigui ratione ita repraeseulare/ heel : , , .
_i.- ' +j. "" '"-"«. -+;■.' '-
siquidem brevitatis causa t = tg X. Fit autem:,
unde am — - — ^ -„ .
sive cum sit: t *- ' ;- - ' ' ■ '- . t,
fit am :=
'^«'«- , ... = 1"°-*" + . -■ ■ +,-
Digitized b,Goo';5le
toi
quae est formtihi 24). E nains difi^reyluliAQii prodil: ^ _..
onde etiam, poaito — <j loco q sen -^. — x loce jt ; '
i+i-
4q*c«a6j[
! + ,•
'+1'
40.
E fonnulis propositis, poaendo x ^ rel aliu modis facile ernantur setjaentes :
-il-+ '-'
1+q ^ »(i+qV. > 5(l+q') .. 7(H-q')
I) lofV
I) -lojk' =
"-T- = ' +
= 1 +
»CI+<1') •ll,+ i')
5(1-,") "^ 7(1-,")
»(»+«. > 5(l+q')
1'-'- :■- *^'-^ ■ '
1-,'
ir 1 — 1 l-q» T 1— q'
T+r* l+q' * li+q' * '
Hq-
! + ,■ ^ l + q.
,4+1
, «q
'1+?^
jq*-- . ' j 4,*"
+-^-.^11;=-:-^.
«q'
l+q-
'q
«q'
l+q-
, «<•
d+ql'
<+q"
• l+q" :
, gig!:
,, Google
t04
4kkKK
16, . »8,' , »!,■ .
=
i«q(i+«') , iSqHi+f) . i«i'C>+.0
(i_,.). 1 (1-,.). 1 ■ (i_,-)-
4k'k'KK
8, «,■ «,■
Kir
•+s i+q' '+q'
«q . »q' «q' .
(i+q-)'* (i+qy ■ (i+qV .' ••
«kk'RK
WW
4/^ u/V , «>/?
1 + , !+,■ ' 1+q'
-
4/~q'a+q) . 4/V(l+q') . 4/7(l+,')
(1+q)" ' (i+qT ' (i+qV
4k'KK
8,- 16,. Hf
iti*
1+,- ■• 1+,. 1+,- 1
8,< 8,. 8,-
(1+qV ' (■+,•)■ (■+,■)■
4kKK
4/T u/V , »/?
(t-q)' "^ (I-,-)' "^ (i_v>' "^•- - :-, -,-
Formulas 4)- IS) duplici motto repraeseDtaTimas ; facile aatem repraesentatio altera ex
altera seqaitur, abi slngaU denonunatores Id seriem erolmatar. Affnotemus adhuc, se-
cundom tlieoremata §. 37 proposita e dnabas ex eamm nomero, 4** et 8", derivari posse
omnes. Fonendo enim Vq loco q, cum abeat K m(t+k) K, snbtraheiHlo e formula 4}
prodit 5); deinde pODendo — q loco q, abit % io: k'K, uade e formnlis 4), 8) pro-
dennt 6), 10); 6) immutata manet. Pouendo q* loco q al>it k'K ^|/lc'K, uode e 6),
10) prodeunt 7), 12). Ex S), 10), quia kk + k'k'=l, prodit b). Ponendo /q*
loco q, abit kK in 2Vk K, node e 9)- prodit IS), fonepilo — q toco t|, abit kKK
Id tkk'KK, unde e IS) prodit 11). Ceteram pro ipso .Modulo t^L Complement eios-
modi series non extare Tidentur. ' ' "
Formulis propositis ad dignitates ipsius q erolntis, obtinemtu:
15) -l0«k* = 8q+ -q' + y S* + yq' + -.,• + -q'« + -_.,» + — qlf . 1 . .
,y Google
105
16) logi^. = 4,_4,. + ^q._4,' + ^q.-^,» + 5,'_4,» + |,.-^," + ...
17) _i|i_ = l+4, + 4q' + 4q'+8q' + 4,« + «4' + 8<l" + 8q" + 4q» + 8," + 4V + ..
19) -^ = 1 — 4q + 4q» + 4q* — 8q' + 4q» — 4q« + 8q» — 8q" + 4q" — »q" + 4q" . . .
K) -i = 1 — 4q» + 4q« + *q' — 8q" + 4q"' — 4q"+8q* — 8q»» + 4q" . . .
a)
= 4/^- i6/V + «/V-w/i' + «/v-«/q^' + 56/V*
= I_8q' + «q* — Mq' + »*q"-48q»+96q"— 64V* + «q"— 10tq" + ...
Qnaram serierom lex et ratio quo melius perspiciatur, denotabimns eas signo sum-
matorio £ tennino earam generali praefixo. Stataamus, p esse numerum impartm, <p (p)
lummam factorum ipaius p. Turn fit :
tt)_logk'»8 £-5MqP
») 1*8*5.4 E.£*LjV_q'p-.r''-q*'-r'--|
Porro sit n nwnenu impar, cuiua factorea primi omn*a formam 4a-f- 1 habent, 1^(0) au-
merua factorum ipsias n; I, m nnmm omDes a asque «l 00: obtinennu:
«)
tkk'KK
»)
4k'KK
K)
4kKK
db, Google
(4iii — i)'n
SI) -i^= «!:*(.)q '
«, JtK!L = ,_4.+(.„'-- "•"+..*(.),''*■"— '•",
«« «J^ = . _ 4 I ♦(.„"*-«'■■ + 4 . *(., ,«'"<•—"" " .
Designante p nmos nnmeram imparem, $(p) sammam factorum ipsiiu p: fit
M) -l^ = l + »J»(,){,P + B,'P + 5,,-P + 5,T + »*'P+..)
«) i^^55 = l + 8!H>(p)(-,P + 3q'l' + »,>»+Sl"+»l"'+-)
P-'
M) = 4£(— I) «(p)/qP
M) ii^=l + 8Iip(,){-,T + »,«P+»,-P + «,"P+5,"P+..)
») iiH.= 4i:»(p)vV-
DemoDstieniiu formulam 27). Invenimps 1):
lo,k= 10,4/V- 7^- +
4l'
1+, ^ sd+ii »a+« '
qnod ponamiu = log 4v q + 4 £ A*^*<j". Sit x nmuerus impar p ^ m ni', e (juovi.<
tennino -^\ — — -, prodit ^^ — , unde constat, tore A"" = 5^ . lam ait x
munenu par = 2'p= s'niDi': e terminis
"<l+q") ^ «»(l+q'") ^ 4m(l+,.-) ^ «n,(l + ,'") 2i„(,+,.'m,
prOTeuit
» ( t 4 « Ji— ^ jT) t'm ■
nude A*"* = ,^ , id quod formnlam propoutam sappeditat.
db, Google
t07
DcmODfttremus formalim SO), lovi-oiiniu
l + 4IA'».f.
Sit B"" nmuerus ^torum ip«u» x, (juifbmum 4iu+l halMnt, C'"' ouDieras factorum,
qui forrnam 4in+S haljeiit, fiicile pelet^ fore A""=B"" — C'"'. Sit x:=s'no\ ita
nt D sit Dtimerus itnpar, caius factores primi omues foratam 4m+l, a Dumerus impar,
cuius fadores primi omnes fbrniam 4 m — 1 haLent, &cile probator, nisi sit n' mimeros
(juadratus, semper fore B"* — C'^* = 0, abi vero u' est Dumerns qnadralua^ (ore B'"'
— C'" = B*"=si^(d), formula 80) fluit.
Fostremo probemiu formulam 84). loveoimns 8);
Designaiitex numerumimparem, facile patet, fore A"" = l^(x) ; nbi vero x uomenu par
= 2'p, desigoantep Dmnerum imparem, (jaoties m factor ipsins p, e termiois
prodit 8mq"{l — 2 — 4 — 8 — .. — 2'"' -t- 2')= 24mq*, aiide eo casu A"" = S$(p),
id quod formulam proposilam suggerit. Beliquae similiter demoDstrantnr vel ex his de-
doci possuut.
^ tXi . tKi i J J- . . .
Expressiones cos am —^, A am , — — ad djgmtates ipsius x evo-
lutas, Coiitficienlem ipsius X* Danciscimur reap. TlT')' r("^~"/' "*~ "5"f~)*i
uadee formulis $' antecedentis Zt), SO), 24) prodire Tidemns sequentes;
"' M-r^;- *»!+, + ■+,. + 1+,. + !+,■ + •;
I (1-1)' d-q')' d-q')' "I
'" M^ ^ ( i+q t+q- ^ i+q- i+q' * '
. . .( q(l-«q'+q') q'(l-»q'+q') , q'(l-»q'+q'1 (
■^ I (i+qV (i+q')' (i+l')' ■■)
O 2
Digitized b, Google
loa
I (i-«. (!_,•). "^ (i-q-)' "^ ■)■
fix liu posito — (J loco q obtiaemus :
V « ; I t — <I 1 — q' ^ 1— q' 1 — q' ^ (
\ " / I 1— q 1— q* 1— q» i— q' /
45) kkk^iiLV^ «(^ i3!_+-B: ^3!_ + ..J.
V * / I i+q' i+<t ^ i+q* i+q* ^ /
Formulis 42), 44) additis, obtinemiu (-^j ; 4o)et48), il) et 46) subductls obtiDemus
/ SkK I* / tk'K V ., .^ / — I, ,.. /«/Tki> /«/Vkv
I 1, I 1, e quiuus posito resp. ¥ q , q loco q prodit f — i 1, 1— I 1; e
/■♦/^KV ., , 1^ ^ /4/kl?KV
I — - — I posito — q loco q obtmettir I 1 -
Sub fiDem, posito k^sio-S-, erolvamus ipsam '&=Arc. sink. Vidimus, po-
sito /q^ !oco q abire k' io ^~ -p » pODamas rursas — q loco (j , abit k in ^ , sifc Id
i. tang -9-; ita nt posito ivqloco q, expressio ~ °f — matetur in
Hinc e fonnnla s)
- log k' =
8,
+ X1-,
T + 5(1-,-!
1 ■'■'
l-I-
+ 7(1_,"
) '
4/T
i+q
4/V
4/7
+
quae in hanc &cile transfbrmatur;
47) — = Arc tg Z^— Arc Ig /V + Are !§ /V — Arc Ig VV H
quae inter fbnnnlaa elegantiasiinas censeri debet.
d by Google
lOD
41.
Aequatiouem supra exkibitam:
„n ■m = ; h — — 1- . - -
w ir 1 — q 1 — q' 1 — q*
in se ipsam ducamaa. Loco 2 sin m x sin n x ubiijue snbstiluto cos (m — n)x — cos
(io+n)x, factum indait foimam:
I j Mii*»n — = A •!- A'ctuti 4- A'coiii +A''cm6i + ....
li)Tenitiir :
(!-«■ (■-<rt' (!-?•)■ ■
Pom) fit:
A*"' = 16B'>" — 8C'"> = 8[2B<"> _ C'"»j.
siuoidem ponitur:
C""
q'
lam cam sit:
1'" 1° I q" q*"'" 1
(i-q")(i-q'"*") i-q*" I i-q" i_q"*
St B'"' =
q' I 1 J. q' 4. 1' J.
1"" q"
+ -
T^^+l
l — ^n 1 l_^i> + » ^ ! — ,">*' ^ I — q«n
sire sablatis, qui se destmant, tenninis:
q-—
■>' /_;_ + _£- + ..+ '•— I.
,, Google
110
Porro fil:
nnde
Hinc tandem pnxlit:
A"< = a(iB'-- c<"'l = -'"f .
t J 1 — q*"
unde iam:
Simili modo vel ex 1} iDvenitiir;
J, j_jco.-„— - = B + 8J--j--;p-+-^--^j-+-j-;j^+..(,
siijoideDi :
E nolo Calculi Integralis theoremate fit, quoties
^1 = A + A'eo>Xx-^A''cM4i-h A''cM6i-f- ...
tarminiu prinms seu constaos:
A = -iy^ ' *w-d-.
unde oanciscimnr hoc loco:
B = 1 — 1 / C4M'atn « .di.
« \ « /y It
Digitized by Google
Ponnnuu cum CI. Legmdre
SK tK <K SB
HiDC ettam, cum niutato q in — q, abeat A itt — B, K id k'K, sequitur simul abire
E" in ^.
Adnotemua adhnc e fommla l) sequi:
I (l— !)• ^ (l-qV "^ (l-l'l- "^ "/•
uode etiam mutato q in — 4:
., ..w.fi5.)-= «{_JL^ _ ^ + JliL _ •131. + . . 1
I IT ; I 1— q* 1 — c|* 1 — q« 1— q' /
" ( a+q)« a+q')' U-q'r "^■'C
Subtjacta fonniila 4) a 8), procUt:
-,( ■f('+*q'+l') , 1'(> + «1' + 1") . q"(l + «q"+q») (
= '"1 (1-qV + n-qT + (Trq-::? + •)■
qoem edam e S), mutato q id q*j obtines-
dbyGoot^Ie
lU
42.
Methodo simili ati^ue formala 1) inreDta est, iu expressionem
in seriem evolrendam inquirere possemus, siqaidem formula 18) $. 39 in se ipsam du-
catar. Id quod tameo facilius ex ipsa l) aLsolritur consideratione seqneute.
Eteoim formula :
d.log«i..n,i^ jjj y l-(l + kfc)«a»Kn-^ + kkM«,
itenim (liffierentiata ^ factis reducliooibug, obtiQemas:
— 7? (— j{""
SKi
«K. I
lam Tero invenimus §. S9, 6):
iog»iii»m— — = logi
uode
Forro est $.41, l):
«K »K _ «_ gE' _ g ( -qcoigi 8q'cw4. 54'w6x |
,y Google
uode cam e formula l) sit:
I 1 d' IOC HO am ~
- . «Kx I « / « d»'
proTeoit, quod qDaerimos:
JK »K _ ^K_ «E' 1 _ f q'CT*t» t.j*coi4« 8q-co«6» 1
IT JT Tt IT Hn'» ( i — q' I — q* 1 — q" /
Mutatis simul q ia — q et x in — — x, mide K in k' K ^ E in -rr- §■ 41,
tKx . ZKk , . ^. ,..
Sin am in cos am al>it , e 2) prodit :
/ Sk'K \' tK »E' 1_ gi q'co»l» _ *q*co>4« gq- co.fi « _ i
His adiuDgo, qaae facile e §. 41. l) nequuntur, hasce:
V w / ir IT ft I 1 — q I — q» I — I
(^)'
gK 8E „ ( qco»t« gq'cwii , Sq>Goi6i
"^'"i;: ( i — q' 1 — q« '* I — q"
quamm 6) e 4) sequitaf , mutato x in — x seu q in — q.
Fostto y^sinam— ^, r (1 — j-y)(l — k'yy) ^ R, fit;
^= -(")■'" + "-"■'■>
P
Digitized b, Google
114
-^ = (-^)7» + »"k + k* - 60k' (i+k-),' + i20kv)R
cet.
cet. ,
ttix _ gKx _ (l+k'W 8K» .' l-t-Hk'+k* / «k» 1'
ideoque :
(^)"
-^ + (-^)(^/+^-^^(^/" + -
qua formula comparata com S), eruitor:
(-^)(";=4-+(")'-"-^
' !-,• ^ 1-,. + I-,. + >_,. + • ■ -
H(^)-(,_.H^.iil
*.9.*
Forro fit:
sive cnm sit : 1 5 = 2 . 8* — 1 :
16
,■ g,- »-,• «■,• 1
De luc formala detndiattir seqneos §. 41. 3):
k.fiI.V= i6(_J!_ + -£3L + Jl2L + _£i.
"I, , ; ii-?' + !-,• + 1-1' + I-,'
Digitized by
Google
115
fit resuluuDi:
7, fi£ii)-= , _ is'-J !l3!_+_^3: s^+..),
' \ - I 1 1-, i-s' ^ i-q- 1-^ ^ ;•
unde eliam, mulato q in — q:
\ « y ^ I l+q ^ l-q- ^ l+V ^ i-q- ^ )•
quae difficiliores indagatu erant formulae. Qoas si iis inagis, qoas supra iuTeaimos.
lam qualuor pnmas dignitates ipsoram — ■ , m series satu concinnas erolutas
F0R91ULAE GENERALES FRO FUNCTIONIBUS
. , iKt I
IN SERIES EVOLVENDIS, SECDNDUM SINUS VEL COSINDS
MUI.TIPLORUM irSIUS x PROGREDIENTES.
43.
Iiiveiiti.>i erolalionibas functionum :
iam quarslio se offert tie eTolutionllius alriorum dignitahim ipsius
perageailis. Facilis quidem in Trigouonietria Aaalj-tica via constat, qua, evolalione in-
veota ipsorum sinx, cosx, progredi possia ad evolutioiiem expresstonnm »in''x, cos"x ;
nimiram id succedit formularum notarum ope, qaiba^ 8iu"x, cos"x per sinus vel cosi-
aus multiptonim ipnius x lineariter exhihentur. At in theoria Functionum Elliptica-
P2
d by Google
rum illud deficit sulisidium ; ad aliud confagtemluDi erit , quod in sequeutUms ex-
ponemus.
Formula, quae ex elementis patet:
*'°^™° = iiiiB»-'amu/'l — (i + kk)wii'»mii + kkMB'»mu ,
iterum differeutiala , prodit:
I) "j„'" -— -= "(a— I)"ii''~'»mi' — in'Cl+k*)rin*'«nu+ii{ii+I)k'«i»*'«iiii.
Fosito successive n=l, 8, 5, 7 ••, n = 2, 4, 6, 8.., hinc duplex formetur
aequalionam series:
I.
= — U+^')*ina)nii + Sk'uD'amu
= SuniniD— SCl + kOrin'wDB + Uk'»iii"«mu
= fOnii<3ma — 2S(l+k'}aiii>amu •(- SOk'uD'unu
s= 42iii>t>mii _ 49(I+fc')riii'wnu + 56k'(m*amu
cet. cet.
n.
= I — 4(I+k*)nD'amil -f- 6k'(i>«ainu
:= Uain'amu — 16(l-f-k'>«ia*ain« + tOk'siti"»niu
= SOim'amu — S6a+k")Mii«ainii + tik*«iii*ainu
= 56Mii*an>u — 64 (I ^k') no* am u + 7tk'*in<°ain«
cet. cet.
du»
a' sin' am D
do»
d'iin'amn
du'
d'tiD^ama
d SID am a
d^
d'tin''»mQ
d by Google
117
Ex aequationihus I. eruis successive , posito n d =: 1 . 2 . 3 .
n4. k'iin»wnu = °"'""""" + 10(1+ k*) - " •""*■"" . ^ 3(S+tk'+8l»)«oai
du"
i*
.Miaoiu
du*
<!•
tin am D
du«
d-
■iD^mu
ne. k.«..'.«- = " ■•■";— + »5(i+k-) ''*-'j"'"" + 7(g7+aBk'+s?k') ^'■*^J'""
+ 45{5+»k'+8k»+5k'')iinBmu
08. k'«a'««u = " •""'"'" + M(l+k') '^'■■'"'"" + 4J(47+58k*+47k«) ''*-''°"""
+ 4(Stt9 -4-8015 k' + UlSk«+MS9k*} — -J—
+ 81S^+K>k*+Uk<+S0k<+S5k-)<iDamu
11. a.
n5.k*fiD«»mu = " "'" " ■+ *(l-f-l:^»iii'amo — S
n^. k*»i
d=
sia*>iiiu
d«»
d*
nn'tinn
d«*
d*
liD'amu
m . k'MQ^amu = '-—-^ h Sefl+k") — '—- + l«C7 + 8k*+7k')-
+ l«(18+lSk*-t-lSk44.18k'>}*in'ainu - 48(S4+»k* + S*k*)
cet. cet.
lu videmtis, generaliler poni posse:
S) nJo.k'-Mn'o+'wnu s*
du'"-'
8) n(tn— «).k"'-*«iii"'Mn«
- + <"
j„..- — + °" j^.-. + ». J...-. ■ +■■+«; ""■■■"" + ". • .
(tesigDaotihns Al,'"^ h'^* fiinctiones ipsias kk iategras ratiooales m" ordinis, excepta B^,^",
(juae est (n — 2)*'. Forro e fonnnla, unde profecti samos , geaerali:
'''*^" I = atn— ^Mn^-'smu — iio(l+k*)«o''«niu + o(n+l)k'«i»"+''«inu
Digitized by GoO^^jle
as —-_
patet , lore :
4) a|,"'> = A^^\ + Cn— !)'(!+ k')A;;;;.-" — (Sn— t)*(2n_I)(*ii-8)k"A;,"!.7"
quiLus in-fonnulis, qaoties m>D, poni debet Ag"* = 0, B'""=:0.
Mutato u in u -t- iK' ciun sin am n abeat id -r-: , in formulis propositis loco
t u poni poterit —r-. > unde proveoiunt setjuentes :
d'.— ;
d'.-:-;
dTi'
d*.— :
du»
d».-—
i
J-. . ■
uDimu
du*
a(S+»k'+8fc«)
ac geoeraliter:
nzn
■ + a;".
d—._i-
db, Google
119
44.
Quum inventum sit autecwleotibus, sitjiiidetn ponitur u =: -
per Iiasce:
nn txa — ^_ , On* am ,
earumque differentialia , sectmdom argameatuiii n sen x sumta, lioeariter exprimi posse,
iam ex hanim evolutiombas, secandDin sinus ve\ cosinus maltiplorum ipsins x progre-
dientihas, illarum sponte demanaot.
Ita naociscimar:
e fbromla:
iiL..i..„i^ = .{ 1^"- + /7-i»" + /V"-"
sfitjuentes :
,«IKV., IKi
S.S.4I— ^Inniam— — ei
Digitized b, Google
itO
.{»(i>+.i.-+»k.)(^)'_ 5...o(i+k,(i|l)*+ (.} i^iii +
e formula:
sequeotes :
K IK !e' ,r»<r«»»- 1
1-,. -^
1-,-
-(^r
Mn* am
=
4„+k,(^)'(if
-^)-»-(^r
-•!»■*<'+'■)("/-
-}^
-
-i{...(.+k,(i^J-
-}4^
-
_,J6..(l+k')(^/-
=1-^
-
, , . ./ «kK -i' . . »K.
A,„ -., . B. ,/*K \'/8K «K 2K «e' \ /SK\'
8(8+7k-+8k.)(_j (_._ -.„__]_ «k.(l+k,(ii^)
- «{!. e(8+7k'+8k.)(i!ij- s..»ci+k')(^j|V »} "^'"V
- 4J4. 8(8+7k'+8k<)(i!iy_ 4..K)(t+l')(ilJ+ 4-} _5l^^
- 4;6. 8(8+7k'+8k.)(ilj*- 6..»a+kl(i|iJ+ »} -Sg^
cpt. cet. ;
Digitized b, Google
e formula:
1*1 -
m.
• +
+
4^1
daSt
-1'
sequentes:
•!"+'-'(-T-)"-'')^;^+
db, Google
I
lit
~ /-15- _ .?^' 'i _L ^ _ - / gq'co«gn 4^*co«4» 6q'co.6»
sequentes : - .. '
»io' am (
w I
_,{,
!'-'■+'■'(")■- "I^
-•{•••<'+->(^)'-}^^
-{*-'■+'■>(")■- '-i^^
— (^r-
+ r-. ' • ■' + Md+k-jf-^lL ^2.!_ + _
..{t.8(8+7k-+SV)(^)'_f..»(.+l')(^)'+»'>-i^
Digitized b, Google
US
-4{6.»c8+?k-+8l.)(^)'-6..S0Cl+l')(-yLj'+6'}-t^
45.
Exetnpla antecedentibus proposita doceot, quomodo e formnlis 2), s), 6), 7) §. 4s
erolutioaea functioDom «n" am , „ — inTentaotur. Quanlitates Aj,"', BJ,"",
a quibus itiae pendent, ope formularum 4), 6) ibid, successive eroere licet. At expres-
siones earum generates indagandi quaestioj cnm .nimis illae compjicatae evadaot, qnam
ul eas per inductionem assequi liceat, paollo altiiu eat r^wteade. Onem in finent se-
quentia anlemittimus.
Kola est fonntda elementaris :
, . , , Siinsm v.eftiamu^amu
tw ■m (n + t) — iia wn {o — ») = — - — —-r-, — .
qua iDlt>grata secundum n, prodil:
E theoremate Tajioriano fit:
«iDam(u+T} — >ioaiii(u — t] =
y^du{.iD.m(u + T)_,i„„„(„_,)j
_{. d*.«ia»mtt »> il*.fia*inu t'
'i""""'+— I? nr+— 37 nr + -
0»
db,Goo';5le
144
Facile enim constat, |M)silo u == , et sin am a et generaliler; j l °^ — . evaaescere. Hiac
aeunatio l), etiam altera eius parte evcduta, iuhancabit:
„ , <l* . tia am u t' . d* • wn »"> o v'
" "•■»»•' + — j;j nr + — 3^? m-+- =
Forro aequationiLiu notia:
» {U_t) -
1— k**li]*amuMii*M
in BC dnctia^ obtiiiemus:
S) Mn'am Cu + t) — siii».iiiCu — y) =
4 ^n m n eoi un u A am u . naatay coi am t A am t
'}'
[l — k' Jin' am v nn' am t}* du dv
lotegratioDe fiicta secandum v, provenit:
y^dT(.m' »!. (u + v) - rin' am {u-v)J =
If am n A am u . lin'* am t uu* am t . rf .
1 — k' hq' am u wn' am v (1 — k* lin' am n tin' am 1} d u *
Qoa dcDUO integrata sefnindam altenim elemeulum u, obtinemus :
4)y^duy^dy{«»'am{u + v)_.in*am(n-T)| =
^ ^ ■^ log (1 — k' ««' am « lin' am y) .
E Aeoremate Tajloriana lit:
«ii>*am(ii>|*T) — MB* aita (a — t) ■!
t 4« ■'**' du' ■ ns '*' do* ■ ns +••/'
,y Google
lU
y^dT{ri»'.m(u+») - «B'«mCu-v)j =
•1^^^^-^ + ^
/-./.,{..™,.+.,-
«.• — (.-.)} =
'{-—-^^^'^^^
»« d*.rii)*amu
■ n4 ' du«
n6
-,{„...
-^ + — ^ +
..}.
.+..}
siquidem per cJiaracterem U""*^ ralorem expressionis ^-^ denotamus, quern ob-
tioet poslto n = . Bine aeqnatio 4) ^ edam altera eiiu parte evotata , iu haac a)>it :
«* d'.an'afflu T* d«.un*amH v*
») «»■.... -gj- + J- .^jj- + J-; ■-nr + --
-■»{'""-5r + '"-'-Hr+-}
1 k* k*
— . lia' UB n rin* atn t 4- — . nn* am u lin* *ai t + -r- *>■>* »n ti tin' amr + ,. .
His rite praeparatis, poDatnr:
u a: nn un u '4* ^i ■'"^ Ml n ^ B. ud* un u •(• R, tia' im n ^ . • . ,
«c generaliter:
u" = (mh^ibb + R, Mii>«niu + R, •!«•■»« + B, lin'wnu +...}" =
iiaO*m n + R^"' lin"*' »m n + R'^"'mii"** am it + R'^"'Hn''*' ama + . . - :
porro e rerersione seriei :
u = un im n -(■ I^( ■'A* ■» <> + H* *<t>' am u ^ R, tin^ >m m + . . ■
oriatur faaec:
■in am n = n + .S,ii< •!• S, u> + S, u' + . . ,
d by Google
i86
ac sit ruTsus:
«n» »ni u = {u + S, u» + S, u» + S, u' + . .j" =
lam ex aeqaalione 2):
. + ...=
du' ' ns ^ du* • 115
evolutis V, V*, T*, cet. in series secundum dignitales ipsios sin am t progredienles , in
utraque aequationis parte Coefficientibus elusdem dignitatis sin'"*' am t inter se com-
paratis, provenit:
2n+l
„in . J. T»<" d'.»in«mu , „„, d*.«iiamu d'x.finimu
^ """"" + ""-' m.6a' + *"- n5.da« ■*■•■■♦■ ngn+i)du" ■
Eodem modo e formula 6) provenit:
k"-»iiii"'amu
^ 7Z =
.„ _e^-'»U . R.4. ■)'•"
n« ^ "-» n4.dll'
19.4 ^ 5.6 ^ r.8 ' ' (So— l)2a/
E 6), 7) mulalo u in u -f- iK' setiuilW:
8, i =
(I n ■+■ 1) nn'" * ' •m tt
n^nr + "-' ns.du' + "— ns.du. + • - + d7^
*) Fi( eniin e notalione propoiiti.* i1d' •ran sa u* -^ S" u* 4* S'''n* '^ S'''u* -^ . . , unde cnm lil V"
= . . ■ ■ pr* Talore u s= O, U""" = IlXai . SlT' ■
Ji^m m-i
Digitized by Google
nC.iin'Mnu "^ n4.do' ^ n6.du« T • ■ -r ntn.du'"-'
~ ^ (i:! '•' TT? '•' "tT? ''" ■•■ 4Sd-i)Sii./ ■
Quae sunt formulae, qnas (|iiae«mDiDS, geReralas, quarum ope sia'aoiu, — r^;
e sin am u , sin' am n , — : , — :-; eorumque differentialibus inveniuotur.
AdnotalK) hac occasione, ubi vice versa sio am v, sin' am v, nia' am t, cet. se-
cundum dignitates ipsins x evcivisy e formnlis 2), 0) erui:
10)
n(j.
,+l).J.'
—
=
K
**!■■
un
» +
S
")
.■m'amii
■ +
Sn + l
n(j
.+S)J«'»
(!n+l)(!.+t)
1
2
s;*
'■in'
•IIIU-I-
^C
.tin* •inn
' +
^s::...-™.
+ •
-i^
Paucaadhnc de inventione ipsaram R^', S^' adiieiendaimBt. FomIo siu am u = y,
fit e definitioDe proposita:
=/.
- = r+ii.j' + >'.y' + Rir' + ■
/"(■-T-X'-k-j")
Dmie ;
/(i-rt(i-k'jT
+ »B.,- + s».y + 7B.I- + .
+-^-^l.'+-
S.4.6 t.t't ■*' t ' S.4 "*" 4.4.6
■■»■'■' , '■»■' i|,. + il .i:ik.+ J. iili. + .
i.4.6.8 ^ «.4.6 ■ « ^ ».« f.4 ^ » ■ f.4.6 ^
db, Google
us
■j-(I + k")
-(l+kV-^l"
l.S
S.4.6
l.S. 5. 7
■ l'(I+k')
(l + ky -
41^ .,, + ., ^_-_
cet. eel.
sire etiam:
SB, = i_^. k'k'
58. = I - ^.«k'k' + ^ • k''
9H. = 1— ^.4k'k' + -
cet.
aive deiiitfue:
SB, = fck +-i-. k'k'
l.S. 5
*.4.6 ■
1.8.5
e.t.6 '
cer.
1
5R, = k« + J-.tk*k"
7R, = k« + -
-.8 k'k"
1.8
l.S
k'«
.— .sk.k-.+
9R. » k' +-L.4k'k"+44.fik«k''
2.4.6
l.S. 9
t.i*
irk** + -
cet.
,y Google
129
Ex bis qaatuor tjuauiiiateti R„ exprimendi modiii, modus secuodus repraeseDUiioQflin earuiu
satis tnemorabilem et concionam nippeditat, liiqaidem inlrodncitor quontitas:
Ilae. g. fit -i^ =
1.S.5
1.X..6 1.X.S.4.S ^ 1.8.S.4 S.4.6 '
qua expressioue sex ncibos secundum r integratis, oLtioemus:
+ Cr« _ C-r' + C",
2.4... IS t.4. 6.8. 10.2 ^ S.4.6.8.2.4 2.4.6.2.4.6
designaDtibos C, C, CT' CoDstautes Arbitraiias. Quibus commode determiualis, prodit:
unde vicissim:
• 2».n6.dr»
eodeniqne modo obtiuetur generaliter:
kP'd^frr— ly"
12) (Sin+1}R
»nnin.dr«'
Conferatur CommeiitatinDCuIa (Creil* lournai V. II. p. 22S) iuscripta;
jjUeber eioe besoudere Gattung a1gd>raischer Fonctionen, die aiu der Eut-
1
,,-wicklnDg der Function (I — £xz + z') entstehn."
Inventis quautitatibus it„, per Algorithmos nolos perreaitnr ad eruendas quautitates B*"',
S^' eas, ut sit:
[i + B,. + R,«' + R,.' + ,.1" -I +r^'". + r;"'»> + r;'".i + ...
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ISO
pono uLi poBitnp:
y -. {I + B,. + B,.' + ",-'+■■). .
■" = j"{i + s;"'y + 5;"'y" + s;"',> + . .),
quae cum defiuitioue (jnantilatum R^% S^' supra proposita conveuiunt. Fit aatem,
posito :
♦ W = i +«.- + a..' + H,,> + .,..
e theorematis a Gl. Maclaurin et Lagrange inventu:
"'"' =
8U|iii<Iem traasaclis difierpnfiatioDiLus pDottur x ^ .
46.
Formujaruin (i), 7)^ 8), 9), $. 46 beneJlcio naQciscimorevolutiones gencrales:
^"T"'.i."".., '"'
"(^/"--^(^)"-+-%^(^r
^ (-1)° 1 /Tii-
n(Sii+i)/ 1-,
n(!iH-i)/
Digilizedb, Google
.B". /ill
Ul
"c.(^r-
*» in
_ . '-\ ' I '"'V ' / J. J. (-ir«— I <fv«e.
\ ns n4 T .-r jjj^ / !_(.
(Jn+l) «»•" + ' .in-
..(iLi- B-.fiir-v.-j-
•{-^1^)""
*{«:"(^)"
(^)"
f—iy>yt, j ,'«io8:
^^— :: — 11.
n(«.+i) ( 1-,.
"r""-l— J (— ^r)-^\,) i».« + 5.6 +7:? + -+(i._i)!.; +
--■(^J- -:^.(^J""
(_iy.-.y— . I ,■„
R 2
,, Google
ISX
.{W '-•-■(^J-
1-1"
( ni "" ni -r ■ -r ^^^ ^ i_^,
E formulis 6), 7), 8), 9) §. 46 aliae dedocx possunt, tjuae respeclu fuoctio-
nntn COS am D, tang am u, Aamu eastern partes aguat, quam illae res[>ecta fauctio-
Du sin am a. Etenim e formnla :
nude etiam:
■in *in|k'(K — u>, ^j = CM am n,
videmos , in formnlis propositis , obi pooitur -^ loco k el k' (K — u) loco u , abire sid am u
in cos am n, nude inTeuiaator lunules formulae, (jnae ip.u cos am n respondent. Fon-o
ex aequatione:
■in am i n s= i tang un (a , k*)
patet, simni mntari posse u in ia, k in k', sin am h in tang am a; nnde formulas pro
tang am u emimns. Ex his deinde , tjuia
cotang am (u^-iK'j = i A am (u),
formulas [nt> A am u emere licet, cjuae lormulis 6) - 9) §. 45 respondent. (.>iiitius in-
rentis, methodo plane similt ex evolutionilms fimctionnm:
fK» , 1K» . £Kt ., tKx
CO! am ■ , cw am ■ , a am ■ , A' am -■ — '
SKi ' , 2Kx ' . ZKi ' .. tKx '
coaam coi am ' i Aam ■ ■ " a* am
a nobis propositis, evolntioiles generales dtsducis funclionnm:
coa" am ^-^— , A" am .
« It
Qaae snfficiat addigitasse.
Digitized by GoO^^le
1«8
Tratuformationes insigoes stfieram, in quas FuDctioae:i EllipUcas eToIviauiSj uau-
ciscimor, posito ix loco x el adhibitis formali3, quns de reductione argutneQlt imaginarii
ad argamentum reale ia primis fandameatis dedimus. Quae vero cum in promlu siDt,
hoc loco diutius his Qolumus inimorari. \
IKTEGRALIUM ELLIPTICORUM SECUNDA SPECIES IN SERIES
EVOLVITUR.
47.
lotegrata formula supra exhibita $.41. 1):
/2kK\*.. *K< tK tK SK tE' .iCqtoiSx , *a'cot4x . 6q<co<Gi , )
I tr ; > * « ir ir I 1-q' ^ 1-q* ^ 1-q" ^ >'
inde a X =:: usque ad x = x, proTcnit :
1 tK tK _ ^K_ tE* I _ < qwnt. q'»in4i q'»in6i q«jing _« 1
I ir w ir n f \ 1— q* 1 — q« I — q" 1 — q« ■■•^■
Desiguemus in sequenlibus per characterem : ■ Z | — ^ I rxpressiononi :
I qiiAti q'«in*» q'wnfii q'rioSi i
E CI' Legendre uotatione crit, posito =u, (p=:am»:
f'e(»)-e*F(<P)
f) Z(») .
FuuclioDem Z (a) loco ipsius E (ij>) iu Aaalj'sio Funclionum Ellipticanmi iiitro-
dacere coarenit; qnam ceterom ope formulae z) ad tPrmiuos Ci" Legendre usitelus re-
vocare in promta est. Adombremns paacis , (jaomodo ex ipsa erolulione fuitctio-
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uis Z, quam formula l) suppedilat, complares eiu* proprielates » ersi Dotas, deri-
Tore liceat.
Mutelur in l)xinx+— , prodit:
uiule ;
•i£'z/i!^)-ig-z/iiil^.K) = 8i ■»"■'■ + ■>'"''■ + ■''■■'°' +..}.
ir. V«/ «Vir / I 1 — q' 1 — q" 1 — q" /
Porro mutelur in 1) x iu Sx, q in q*, simultjue k in k'", K in R'", prodit: ,
nude :
Al supra inrenimus:
iiiL.;„.„ Hi. = </ '^•'" + /Ei^ /T"°«- + . I
uude mutato q in q', x in 2x:
I « ' ; I 1-q' ^ l-q« ^ l-q" ^ (
HInc se((uitur:
„ tK („/tKx\ „/tS.x „\i 4k">K'*' , /4K'*'« .,\
" — in-T-) - ^{-ir- + ")! - -^r-"""(-^r-' '"')
4, Hz(i^)- l£:z(i£li.v.-.)= ""7" ■i..m(ii^. I.")
In (|iiibiis forinuli.s, qnnrum 4), 6) Iramformalionem functioois Z secuadi ord«nu sup-
pedttfliit, est:
,, Google
IW
Rti de traDsformatioDe secandi ordinis, a CI. X^eguiidFe proposita, constat. Unde-tormu-
lam S) ita qucxjae repraeiientare licet, poiulo u = :
6) Z(u} _ Z(u+K) 1= 1^ MBwnn .uncMmn.
FoDamus brevitatis causa: amj — ~, k""'| = (f""*, e formula 4), posito suc-
cessire k'% k'*\ k'*', .. looo k; 2x, 4x, 8x, .. loco x, prodil:
(jaam dt-dit CI. Legeodre formulam.
Simili modo, e formula §' 41:
iE i£ _ iJL ^ = ,/^_ + ^!l_ + _3l_ + ^L + . .1,
(juam etiam huoc in modum erolvere licet:
jrK_«C^ «K JE' ( , 8,- »,■ 4,< )
» « . » I l-q- ^ l-l" ^ !_,• ^ l—q- ^y
comparata cum hac, qaam snpra inreuimus:
prodit :
8) «K(K_ E") = (IK)' + «(V'-K"y + «OC"K«')' + «(k"iK'")' + . .,
ijuae cum ea coDTeDit, ijuam CI. 6au«« dedit in Commcut. Determinatio AttrtictionU
cet. §. 17.
48.
Eadem metliodo, qua §. 41 eruinius erolutionem exprcsaioois I 1 sin' am ,
inqniramos ia expressiouem \ Zf ~)\ in seriem erolrendam. Fonamus
= 8, 'a + A'co»«»+ A"coi4t + A-eoifix + ...j.
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quam expresxioDein proposilam intlaere rldemiu fonnam, dam loco 2 siu 2 m x sin 2 m'x
uliique ponitnr cos (m — ^m') x — cos (m+m') x. Fit primnm:
A = "J* 4. 1* . 1" , 1' ,
(l-qT ^ (l-q*)- (l-i'r ^ d-q')- ^ ' ' ' '
Deimle generaliter olitinemus A"" = 2B'°' — C'"', siqaidem ponilur:
d-q-jd-,'"*-) (l_,.)(l_,""..) (1_,.)(I_,"*.,
-2^: +■
1(1 — (!'"-•)
(l-q-)(l-q'»-') ^ (l-q')(l-q'— ) d-q'— )(l-q'j '
Id singulis hanitn expressiouum terDiinis sabstitnator resp. ;
t"" ^ q' ( q" q'"" i
d-q' )(1 — q""*") I — q"»ll-q" 1 — q'"«"(
1° , 1° / 1" + 1'-° +ij
d-q^d-q'— ") l-q"'ll-q" ^ l_q"- » ^ /■
prodit :
R'n. - 1' f '■' J. 1' J. q* J. 1
,„ , ,..„ ,.». ,....
q' ( q' I q* . q' I , q" i
- i_,-\i-,' ^ i-q- ^ i-q- ■^••T ,_,../
/-.„, _ (°-"q"
I — q'" ^ 1 — q'" ll— q'
undo;
tin> _ sum) _ rin) — _ t" ^q
q-
+ -
+ •
■ +
,..-. ,
i-q*
i-q-"- )
d-q"
f "
1
^
■ +
q-tt+q'")
d-q"")'
His coUcctis, invenitur evolutio qiiaesila:
s'(t+q*)w>«*.
Ci-«i')' ' (i-qV ^"f
IpsutD A =: „ '' ,., -I- ■■■ '* „. -♦- „ ''l,,. -+-... cam etiam hunc in niodum evol-
vere liceat:
i-v >-q" i-q-
db, Google
1«7 -
ioreniiDiu e §. 42. 6):
_._ '!-!M-!(f)(4-)H
Porro aatetn constat esse:
iategrata eaim aeqaatiooe l) a x ^ luqae ad x ;= -^, lermini omnes, praeter primum,
eranescuDt; unde si Q' Legendre uotationiljus uti placet:
'\l S(f) '* = 5 ■
quae est lotegrelis definiti aatiii abstnui fletermiDatio.
INTEGHALIA ELLIPTICA TERTIAE SPECIEI INDEFINITA AD CAStIM
REVOCANTUB DEFIHITUM, IN QUO AMI'LITUDO PARAMETRUH
AEQUAT.
49.
Anteqaam ad tertiam speciem Integralinm Ellipticorum iii serieoi erolvendam ac-
cedamus, panels, quae TJieorlae illomm adiicere contigit, seorsim exponemus, iduue
fere ipsis aigois Claro eios autori nsitatis. Mox idem novis adhibitis denominatioDibun
proponetur.
Froficiacimur a theorematibns quibusdam notis de specie secunda Integraliam EUipti-
corum. Fit : ,
2 lin •m ti • c«* am ■ . a im ■
X fin am ■ . coi am u . A »in u
■in m Co+a) ^ tin am (u— a) e
s
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ande:
■ ■ma . Anna . tin am n . coiunii • Aama
U — k*aia*KBa . lib'amui.*
qua iotegrata formula secundum n, prodit:
I) /*dd.{5in'.Hi(u+a)-5m'«»(u->)} = »'-«'°> • ^'«°- ■ A.m. . ijo'tmu
nti iam supra inrenimus.
Ponatur: amu=ip, ama=3«, am(a+a) = (r, am(u — a) = -&, erit e oota-
tione CI* Legeudre:
k^dii . •in* am u «= F C«) - E{^) ,
nude etiam, com sit F(ff) — F(«) = F((P), F(*)h- F(«)= F(ip):
VJ^A" • «ii'.mCu + .) = F(^) _ EW + E{«)
k'/an . »iii*ain(u— .) =o F(*) — E(3) - E{«).
Hinc aequatio 1) in banc abit:
t) »E(«)-[eW-E(5)}= *\"2.'J2'T^X'^ ■
Commutatis inter se n et a, abit « in $, ^ in — *, .r immutatum manet, unde
ex l) prodit:
sE(« _ (EW+ E(»)) = "••'°';rf'^*.-.i°' ^.
fjua addita aequationi l), proTenit:
S) E(^}^(«)-E(<r)ak'«iD«.«ni9.m>r,
quod est tbeorema de Additione fiinctionis E, a CI. Legendre prolalum, L c. Cap. IX.
pag. 48. c.
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tS9
iDt^nJia fonnae:
tecundum oam, qtnin CI. Lcgeodre ii^stituit, lotegralium ElUpticonuti diatribatioiiem in
species, speciem tertiam coiutitauitt. Quantitatem — k'sin^M, qnam per n desiguat,
Parametnua roc&tj noa in sequeutibus ipsum angolom « JParametrum dioemus. Qtwnim
iDtegralium , multiplicata aeqoatioae Z) per
ikl<P) ~ aw ^ am '
ac iDtegratioiie iiutitnta a 9 =: nsqoe ad (p^<p, qno &cto ipsius a limites eninl : « = «,
(r = <r, ipsiiu '9 limites : &= — «, -9- = -9, expressionem ernimnt seqaentem:
ES).
Am
Facile constat, cum sit E( — (f) := — E((p), «s»e;
J am "J A») ■ J im
— ^
node cum sit:
r 'EM.d, _ /tm . d» _ /"J
n
/ ' ES) . a? / - E(w . ai!> _ /• 'e(<i).jji _ r e(w . d^ _ / •'ec^.j^
■~AP) — "y i(») y iiw) y ^w) .• ^m
_« O
nacti siimiis noTum ac memoramle
THEOBEMA I.
Determinentw anguli $-, a it a, ut ait:
r») + F(.) = EWi E(i>) - r(.) = »0).
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erit:
^
*<a Mi fertm species Ihtegralium EUipticorum, quae ah elementis tribuspen-
det. Modulo^, Amplitudine (p , Parametroa, revocata sit ad speciem pri-
mam et secundam, et Transcendentem novam
J ' am
Am '
quae tantum a duobus elementis pendent omnes,
50.
Ponamus F(a2) = 2F(a), quoties ip = », fit ff = «^, -9-^0, quo igitur casu
e theoremate proposito naDciscimur:
r k'Mi«coi«a«.A.'».d» = Ff lEf 1 - -i- A '^t»)-a»
Qnaedocet formula, in locum Trauscendentis novae substitui posse et Iudc:
qnod est iDtPgrale twtiae speciei definitum, in quo Amplilado Farametrum aequat, quod
igitur et ipsum tantmn a duobns elemeatis peodet, a Modulo k et quantitate ilia, quae
situnl et Parameter est et Amplitudo.
Ponamps 2F(m) = F((p) -h F(«) = F(<r), SF(^) = F((p) — F(«)==FC^),
eril ex l):
tj &.m ^' ^' J (i-k'riDV-^'wK*)
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1 / 'EW.Jy pn,,.™ P t'.iii9eo.«A».».>.d;.
quibus in theoremate, $" autecedeate proposilo, subslitutis formulis, oLtiuenius se-
qoeos
THEOREMA II.
Determinentur anguli at 'd t^<^f u^ ^'^•
F(»)EW - rwEW + rcd)Ec«) +
.rwEW + rcd)Ec«) +
o
- k'naaMi^ia./" '"^'9 -Ay
J jl— k'iui'9. dii'v) A{v)
gtia/brmula Jntegralia tertiae speciei indefinita revocantw ad definita, in
guibus Amplitude Pa^ametrum aequfU^ ideoque quae ab elementU tribu*
pendebantt ad aUa* Transcendentea , quae tantum duobus constant.
Conunatatis iuler se « el $, abit 9 in — -&, v immutatnra niaoet, unde cum
insuper sit:
e theoreDiate I:
Digitized by Google
— m —
obtinemus :
^
Hinc, subduclione facta, prodit:
/k**iD«cota^o( . imffp • dyi /* ^'sin^ivxfiAy . tin' a , d«
{I — k'«n'«..iD'¥)A(i/') y (l-k'«D'v.«i.'«jaC«) ^
n
F{v)E(«)_r(=t)E(v)i
quae docet formula, Integrate tertiae speciei semper reVKfcan posse ad aliud,
in quo, gui erat Parameter, Jit AmpUtudo, quae erat Amplitude, Jit
Parameter.
Ubi ia formula 2) ponitur $ = -^, obtinemus:
/^" L'nin tf rjl1uAK«in'w*dc0 I 1 '
|l_l'.in'...l.>lA(.,) -. "' "--•,,.
O . . , ^. _
Formulae 2), 5) cu*" »i3 conTeiiiuntj quae a CI. Legeadre exhibilae suat Cap. XXIIl.
pag. 1*1. (n'), (Vl-
INTEGRALIA ELLIPTICA TERTIAE SPECIEI IN SEKIEM
EVOLVCNTUR. QUOMODO ILLA PER TRANSCENDENTEM NOVAM Q
COMMODE EXPRIMUNTUR.
51.
E formula: > \
Hii'ani U+^) — «in'»in (»—■*) =
»KA tKA >KA . tK» JK. JS^i
»lt. ,<
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quae ex ekmentift consiat, erainuu integiando:
1) i^y^di.{.ia'«.^C-+A> - .m'.m^(.-A)l =
lam dedinuu §. 41 formulam:
Iw/" « irit «« \ 1 — 9' '*' 1 — q« 1 — q' '7'
UDde:
. I «qc».»(i— A) 4i|'i».4t.— A) 6,'co.6(.— A) i
I 1-1- "^ !-,• "^ !-,• "^1
_ , ( «qcoi«(. + A) >,'a»4(T+A) 6q'<ai.6fr + A) 1
is,«.«a™j,
"1 I-,-
. + i£l
im4A.m4i , 6q'sid6A.lii6< ,
1
J
inc fit Kt I):
''^■■
Sk'iinii
tKA SKA . 2RA
ir K ff
«K.
1 l.'^.'™**'^^™.™*'^"
\ I — q 1 — q* 1 — q*
r ,«n«(.+A)
1 l-q-
q..:c4(,+A, q.™6(. + A) 1
^ l-q. ' I-q- ^ 1
( qsitfSAcoiXi
I l-q-
, <i'dn4AM«4>i , q>tiD6Aeor6> . (
ubi ita detemunari debet Conetant, at expressio proposita pro x := evanescat, node
e i. 47. 1):
.( q.»«A , q-».4A q'J.CA- 1 _ , '" ; C'"* ^
( 1_," ^ l-,< "^ l-q- ^ ■ '( ' « V « ;/
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144
Formula 2) a x = usijiie ad x = -j- iotegrata , cam prodeat -j- . Const. , reliqms eva-
(|aod idem est atque S) §. 50.
Designabimos in sequenlibus per characterem n (a « a , k) seu brerios per TI (n, a)
integrale: n(u, a)*) =
u V ,
/k'nntmacoinnai&aina . iln'ainu -da __ /~ k'tiagic«tac ^« . itn^ . dy
1 — k'lia'amu . Ai^tp J jl^k'iin'ac . Hn*y|&(9)
liuaidem (^ = am a , « = am a . Qailius posttis , aequatione 2) rursua integrala a x :=
usqae ad x = x, prodit:
-^^(-irj- > — 1:17 — + »«-,.) + 5(1-,.) +■■)'
qCM»(. + A) VtMt[. + A) q'CMtfr + A) _
■•■ »_,■ "^ «(l-q') "^ Sd-q-) ■^■•
IKx „ /*I^A'\ , f q»in8A»ing« q'tiaiAwil*! q'»io6Aiin6« 1
-T- ^ VT-) - * I 1^? + 8(1 -S') ■•" 8(1 -qV + ■ 7 •
uuae est Integralis Elliptic! tertiae specie! erolntio quaesila.
TJbi aduotatur evolutio nota:
•. »( _ . q*co»4it q*cas6x q*col8i , 1
- |,^Cl_«q«.«.+q*) = t(qco.Ii + X-^ + -3-^ + -3-^ + . . •}.
r <iv
•l Cl' Lcnndra nmllo alia «*t denolatia ; pooil «iin ille n (o , «p) =» / — ■— - ■' ■Trr/ 'r i '•■ "' ■ Vx^ ■
' ' "^ "^ '^' y (1 + n Mil* vJ A (»>
bii nt n(ii, >), ilK erit:
^ : n(— k'rin'K. v)-
Mgao multiplicatorio II, ncpiui a oobli adbibilo, commulelur , *ix i
yGoogte
videmas formolam 3), siogulu evolulis deaominatoribos l-^ti*, 1 — q*, t— q*, cet,
hanc induere ibrmam:
„ n(i5L,iii| =
52.
lategrata fbrnmla 1) $. 47:
a x=:0 usqae ad x^x, prtxltt:
abi Conatant ita detemuDala , at pro x =: eranescat, fit =
''uhf + T(I^ + 7(1^ + ••) = - '-»■-*<'-«<'-'« ■ ■>•■
ideoqae :
n /%/«I-\ . . ( (1 — ^ l^I^l■».^-q■)(l-«q^co^^■ + ^^)(l-«VCTll. + l|") . . . )
' ";?V \-r)-*'^'°'\ i(i_,o(>-i>)(i-«---r r
DesignaliinHU in postennn per characterem © (a) expressiooeni :
e(.) _ »(0).»
designante 6(0) Constantem, qaam adhac inJeteniunataM relinquimm, dam commodam
eios detenninationem infra obtinebimus ; erit ex 1);
* »ro l{i-i)(i-q')!i-q').-.l-
T
Digitized b, Google
• — u«
nmle formula 4) §. 6\ in hand abit:
n(J^.iMj=i^z(l^) + .l.l.,.. *(' " *')
SKi SKA
sive, rursus posito -* ^^ u, ■ ^ a:
K"C.+^)
siqnidem ponitur: = O'(ii)- Quae eat commoda expressio Integralis Klliptici n
per TraDsceodentetD Dovani 6.
Facile constat, asse e(-^tl) ^ 0(a), UDde dMhmatatis inter ae a et u, e s)
prodil :
qaibus a 8) mlKdietu, fit:
*i n(». .) _ n(t. «) = uZ{.) - »z(u),
quae eadem est atque formala 2) §. 50. HiDC, posito n(K, a) = n'(a)j eTanescenle
n(a, K), Z(K), fit:
n«Wi=KZ(.).
quae est 0' Legendre, quara supra exhiljuirans S) §. 50, formula.
Posito iic=a, e S) fit:
Videmus igitur, Transcendentem noTam sive per Integrate / — Tv > definiri posse ope
formulae :
9W
/--'•= /-^
EtTl— E'F(t.)
...0
sire per lotegrale definitum teniae speciei ope formulae:
8(1.1 «.zw-in(., ■)
db, Google
147
E formula S) DfiociBcimar:
_^..(4i)+„(4i.4i),
nude S) in haoc abit fonnolani:
«) n(„,.) = .Z(.) + i=i-.z(^)--i±i-z(^)
que rat pro reductione lategralis t. »p. indeSniti nd definila , ilque cum Theor. II.
§. SO. cOBveoit.
COBOLLARIUM.
Uti iom sapra ex erolutioiiibiis inreDtis Algoritlimos ad computam idofieos dedaxi>
mus miDOs ut Dora proferaDtur, qaam quo melius earnm perspiciatur natara; idem rur-
sos agamus de inventa erolutioue fuactionis
..d.
»m
1(1— Od-lKl-l'l---!
Quern ia finem antemittamos sequaulia.
Fonatur prpductnm infiuitum;
-(4^)(4^)"(4^r(4if)' ••
siquidem iteratia Ticibus »ubatitnitur:,
prodit:
-(-')(4^)'(4ff)'(4^)'(4Tf)'-
T t
Digitized b, Google
148 -
= ('-')('-')'(S-3'(4^)*(4^)
ande videmua, fore:
1) T = (l-q){I— q)S(l — T)»a— q)*(l-q)A... = (1— q)*.
Sire etiun cnm sit:
-('-)(4^)*(4ffr(4f^)'-
fit T = (l— q)/T, nnde T=(l— q)'.
Ex 1) St:
^ .-. = (4^)'(^)'(4^);-
ijua in fonnola loco q saccessiTe pouamns cj, t\'i q\ q', . . , et instituamas infinitam
mnltiplicatioDein. Advocata formula supra exhibila:
•/v=(4f^)(4^)(4^)(^)-.
prodit:
aiqaidem designaiuus, ut supra per k'"'' qnaotitatem, quae eodem modo a q" peudet atqae
k' a q, lire Com plemenlum' Moduli per transformatioDem primam o^ ordinis eruH.
Forro inrenimtis $, S6:
;(I_^(l_q.)(l_,.)(I_,., . .( = .
lam:
Digitized -by Google
148
Posito 111=1, n = l^} " ^ ■ - = m", /mn=iii'; - f — =a f!', /m'i»' = u",
cet.; Qotum est fieri k'^ss— r, k'*''^-T* k'*,''s=-?a, OBt.,' anSe:
Hinc etiam flait, desigoante /« = -7=- limitem commuaem, ad quem (juaotitates m'*",
n'** coDTergnnt:
If 16 m a S m' 8 n" 8 ni'* I
Zju I mn — dh S n « b K d J
quae formulae compatum expeditusimom suppeditant. Docet 5), qnomodo ex eadem
quantitatam serie, qaam ad iuTeaiendain ralorem fonctioius K oalcnlatam haba« delws,
ipaias etiam K' valor confestim proreniat
Formnlam s) traosfonnemns. Fit, at Dotnni est:
Hinc obtinemus, siquidem iteratis ricibm simul loco k sabstituimas k*^ atqae radicero
qnadraticain extrahimns:
16k- I / t lev I
( i6k"' ; \ I \ i8k">' ;
i,.,k,., ,1
{^^}*{-'}'-{^
oode posito p s= 2*":
— irf
K Ik'P'k'P'l'l
Hino vidonus e fonnnla 8), q = e limiton fore expressionu { — jg — > , cre-
scente m sea p in iafinitnin, qnod est theorema a O" Legendre inventam.
Digitized by GoO^^le
i»«0
Netf pon rel ipso ifltuita formulae a nobis exhibttae>:
I.. ,,r^l (■-H'K'.+T')('+q')('+l-) ■ ■ .' l'
( (l+l)!l+l'J(l+«')(i+l')..-) ■ " .
patet, neglectis qaanlitatibus ordiois cj'', fore:
■> = y -Ts--
qumi cum (licto theoremate conreait.
lam in formula nostra
loco q substitaamua successiTe duplicein qnaotitatum senem:
et infinitam inatituamns mnltipIicatioDem. Adrocelur formula $*
(1 — llCll.«i+<|1(l — tq'o.lH-yC— '<'P»«»+,' l")-
/f (l+Sl«"««+q')(i+«'l'«»s<+V)<l+«q"i».«>+H|") ■ . ■ •
ac deaifuemns per A'^* expressionem
«_ (1 — tqyco»*pi-t-q*P)(l— tq'Pc<n«pi+q'P)<l — tq'Pccntp« + q"P) . . .
/^;p7 ^ (i+*q''c«»sp»+q''')CH-«q'P«>.«p>+q«P){i+*q'i'«o.«,i+q»P) ... •
provenit ;
AlA<..*,iU>la...ft ^ (l-iqca>t. + qTfl-»q'co.t.+^)<t-tq»co*«.+ q"} ■ . i
Factorem constantem , (|nein adiecimiis, ^ ^_ '" ^^ 7 il■ ' «»n__ n '> ?* ""P^ ii»*«6tia *ive
eo detenmnaTimiis , quod utraqae expressio, posito x=0,. unitati aequalis eradat.
lam Tero iDTenimiu: •■■■ . ^ .. .
^\~^) a-«qc0.t,+q')(l-tq'cM»« + q')(l~iq'co.t..hq"}..,.
e(0) ™ }{i-q)(i-q')Ci-q')---r
node
Digitized Isy Google
— — 151
HiDc posito — -^ss a, '6m II -= <p, fit adrocalia formulu, (|iias CI. Legendre
de traasfbrmatioDe secanJi ordfnis proposoit, nanciscimnr seqaens, qu^d computum
expeditam fuDctioDiB 6 soppedilat ,
THEOREMA.
Ponatnr am (u) = (p, di=:1, n = k', A(iJ)) =; v mmcos'^ + nnsin^f = A,
et calcnletar seriea qnantitatum :
n' = V mn , »' =a y m'a' ,■■"«« V»
A' =
erit;
■ /'y'E{y)-E'PW ^ .
Cuios theorematis aluqae erolationatn ooBBidcratjoiiE per famuias DOtas ac Jiai-
tas (temoiutraQdi negotio^ cum in promptu sit, siiperscdemiis.
DE ADDITIONE ARGU9IENT0RD1I ET FARAUETRl £T AUFLITIIDINIS
IN TERTIA SrECIE INTEGRALIUM ELLIPTICORUM.
..;.•.- -^ ■ . 5a-
FonDulam io Aoalysi Fanclionis 6 fintdamennilem , et cuius wAm in seqnen-
tibta freqncbtisflKnas tuns erit, nancisciMw consideratioae sequente. Etenhn (juie po-
sit um e5t:
X* k*iiii*inacoiama Aaini • lin'ainii . dn
V 1 — k**lii*ama , «iii*ama
fit:
d« 1 — k*Mi*anta . ^'am'u
db, Google
132
Qna formula secnndam a integrata ab a == lu^tp ad a = a, prodit :
1) /d«. ■ ''"f"' .. ^ . a l-logCl — k'»hi*»m«mi*ann«>.
. ■ . ,
Fit autem e 8) §. 52:
unde :
quibos snbstitntis, dam a logaridimis ad nvmer^s tianis, e. l) t>l>tineii>: ^i i
Formnlam S) ita repraesenlare possamas:
k».
MiiiniacMaina
Aims .
■in*
•mn
l-k'rin'«
»a«n*.
tDU
Dnde commatatis
a et n:
V
lAamn
.«n'
••in«
= 8« + .Lz(u_.) i ili(.+.).
■ . . , ^-j ! Z(u) _Z(u — .) _-^Z{u + a),
1 — k'Mn'aminn'amn t j v -r /•
qnilms addids fiormnlis prodit:
4) Z(n) -I- Z(>) — Z(ii4-a) = k^Moamn . linania . Mnain(u+a),
qnae est pro Additione fiiDctionis Z, atqne conTenit com formula 8) §. 49:
E (V) •)■ E («) — E (r) IB k* un y . (ID a . nn «- .
: , prodit e 4) :
5) Z(n) — Z(if{-IQ e k'«iaamu . uncoamu, .■'•:.■
quam §.47' ex erolntioiie ipaios Z deiirATimns. Fosito — n loco'B, K. — n — - r
B formula 5) obtinemus:
6) Z{u) + ZW=k'«ianm..hi««T. ,
PoMtou=T = 4-> *»• 22(4-) =»-'^'*)- " '
•> i^»*-^.-.-4 = /;5. -a-L./j^. ^^^./ir. ....-I---^.
Digitized by Google
Ids
ForBiulani 6) ind« a n = vsqne ad n ^ n integremns. Cam >it /z (u) . d u =
H-^f prodit:
1.. *•
-loj
e(K)
= _lo8a.n,u.
lire;
»<;+-' =^™
Positt
u
= — K,
ernimiu e 7)
valorem
ipsias
•> So-
°7r
uode
T)
fomtam induit.
„ 8(. + K) _
" 9. =
Fonnalam 9) ex ioTenta evolarione:
\ ^ / _ (1 — «qcoi«» + q')(l— t<l*ewt»+q«)[l— fn'c<nll+q") .
»(0) W-qja-qld-q')...)-
bcile confirmamns. Fit enim, mntato x in x -i-—:
I « / (lH-«qc.«..)-q1(l+»q»»»t, + ,-)(l.).t,.a.t,.|-,-) . . .
em l(l-q)(i-q')(l-q') . . .P •
unde:
' ' •' .;■ ('+«q''"»'+q')('+'q'"'''+q')Ct+«q'»°>««+q")...
g/ "I' \ (1— SqeoiJi+q-JCl— «q'«.i«i+q^CI— tq'e-l.+q")... '
qaam ipsam expTMsionon ioTeoimtu §. SS. = r::^ — ^ nti debet
E formula 9) exfMressiones n(n+K, a), n(n, a+K) statim ad ipsum n(u, a)
rerocaraos. Fit emnli.:
Google
iSi
»)) n(«+K. .)=(»+l[)Z(.)+.
e(m+K-.)
-n(u,,) +K.Z(.) + — log.
11) n(a, .+K)-.Z(.+K)+-1-I««
^■"(— )
(1—1') _
»Z(i) — k'm.ini . ■iaco.m. . 11 + —loj-—_—i + _-!„,.
8(u+.+K)
(ii + .) ^ « " i.«l(li + .)
54.
E formula fuodameutali, cuius ope functio II per fiuuAuioes Z, @ defiuitor:
I) n(..., = „Z(., + -l-lo,.-|g=±,
advocalis sequeutibus et ipsls in Analysi functiouam Z , @ fuudamentalllius :
il) Zu — Z(a-f-*) 9= k'dnaina. MOimn . itnani(ii-^a)
iam facile formulas obtines et pro exprimenno n(uH-V} a) per n(a, a), II(v, a), quod
vocabimus de Additione jirgumenti Amplitudinu , et pro cxprimeodD fl(u, Q-^b) per
n(u, a), n(ii, b), quod Torabimus d« ^(ft^io/M j:/r^u/n«nf( Pof-ame/ri tbeorema. Quein
in finem aduolamus se<{aeDtia.
E formulia:
n,.., = . «.z.+4-W.-|i=^
n(T,.)— v.z. +-i-|,g. ^''— ')
n("+T, .) - (M-»)Z. + t'°*' „>.T-.T-.,
setiiiitur;
9 (■+,-.)
, Google
IB*
ExpressioDem sob sigDO logorithmico coDtentftm:
8(.-i) . »(T-.) . 9(.+,+.)
S(a+.). »(,+.). «(•+.-«
ope theorematis fuDdamentalis m. duplici ralione ad Innctlones ellipticaa revocarc licet.
Fit enim ex eo primom :
»(■+•)•»('+>) =
9(. + .-i)». =
e(.+T+.)8.
qnamm fomuilanitii prima et qnarta in se dnctia ac per secandaiu et tertiam diWsisi
proveDit:
8(.-.).»(T-.).9fii+r+.),
' «(«+.). e(T+.).9(u+T-.)
Sic etiam, qtiae est altera ratio, obi theorema faodametitale III. htinc in mo-
dnm repraeseiitas:
1 8».9.)'_ 9(.+T)8(.-.)
fit:
I 9(.-.)8(.-.) ]'
X 90 I
I e(m+.) 8(T+.) 1'
I 81 . 9(.+— .) >■
* 90 )
1— k'*in'in>n.
""■•""
9(.-,)
. 9(a+T_!.)
1— k»riB'«m(u
,-.)..!.■.„(.-.)
9(._r)
.®(i+'+«>)
i_k'.i.-,m(u+.).™-.m(,+.)
9(o+»)
. 9(.+.-t.)
U 2
, io;";'-! ..Google
iW
( e..8(ii+T+.) i' e(a+T).e{u+T+n)
\ 90 i I — k'iin'ama.«ii»'ain(u+v+») '
<juamm formnlanuu rursus prima et qaaria ia «e ducUK. ac per Kecundam et tertiam ^-
visis, extractisque radicibus proreoit:
e(u-») ■ e[t-») ■ a(u+T+») ^
^ {l-k'«n'.m{u+»).«m'»in(Y+.)l(l-k'riD'>ma..m'»m(o + v-»])-
»' (l-k'«u'Mn{u_^.«ii'>in<T— )((l-k'"«>'w>««.*i'>'»"i("+T+«)I *
Vt ex ipsis elemeiitis cogDOscatur, qaomodo exprassiones S)* S) altera io alte-
ram traosformari posaiot, adnoto seqaeutia.
Uiii in formula, tarn saepios adhiLita:
, , , . , ^ (in'amii — iin*«mv
unaiii(tt-f>T).itDam(a-.T) b ■ r :, . , r-j
1— k'Ho'ainii . sia'amY
loco u, T reap, pooia u-j-v, u — v, prodit:
•in* am (u + v) «- »in* am (a — v)
FoiTo dedunns formnlam :
■■namSa.riowi.., _ - — ..
I — k'iin'wii(u+v) . im'amCu — »)
unde maltipUcatioDe facta, obtiDemns:
ti n • CO! am U-. A«Bi u . mi am i
4) 1— k'iiii*aa>C>>+v) -Hn'sBiCa— v) =
11 — k'»in*»mii)(l — k'«iti*am*l
fl — k'tia'amu .on'umvf
bunSv^l— k*am*amu . lin'.am v f
cnius formulae Iwnefido formulae 2), 8^ iam facile altera in alteram alieunl.
E formula 4) adhnc deduci potest haec generalior:
^ f 1 — k'an'ami) . dn'amTif 1 — k'tin'amn'. tin' am v'}
f 1 — k'nn'amn . Mn'ama ) {1 — k'lin'amv • MD'amv')
(1— k*.ii>'am(M.t.w').riii',m(a — n')|M— V'«a'*'"(T-t-T').Mn'am(T— v')|
^ (1 — k»«o'am(a+Tj . im'am(B — t) J{1-
■ k' rin* am (u'+ **) . ""'•"'{'>'— ■'}!
1 — k*Hn*amo
dbyGoo^^jle
167
At CI. Legeadre eo loco , qao de Additioue ArgumeDti Amplitadiius agit , (Gap. XVI
ComparaUon dea fonctiona Miptiquea de la troiaUma eapice) cam. quae 3ub sigQO loga-
rithmico inveoitur, qQauUtatem sub forma exhibet Kac:
l + k'fiDama. (iaamu . wnam v . 3inBm(Li4- v + a) '
(juaiu noD prime iotaitu palet, quomodo cum expressiooibus a nobis mventis sire s) sire S)
coureoiat. Transformatio satis abstrusa hiinc in modum peragitur.
K formula elementari, cuius frequeDlissimam iam fecimus applicatiooem, fit:
. «naiii(u«HT-
"°'"'(4^)"°''"(V)
1— k'Hn'aB.^-!^±ljtin'>m/l±l - tj
qaihus to se ductiA aequatio^ibus, prodit:
{.-...„(^).,..™(^)} {._.....„(^)....„,(4. - .)} X
<1 — k'unama . linamn . linimv • ■iaain(a^T— •)'
{— "-m— (v)i{'-— m-»(^ -)i
-'•{•■--(^)— -(^)}i-"(^) - --(-^^ -)}■
Altera aeqnationis pars eroluta, terminis
f>e tnulno destmeiitibus , fit:
14-l'*nn**<n| — - — |«in'ain| — ~ — |tia'un| — - — — »)
— k*Mii*ain| — T"—) — k'MD'ini I — — 'liia'aoi | — — — — *| ^
{.-.^«.(^)}{.-....»(^)....(4i-.)}.
. ::-jsijti'Bd=i'Goo';5le
1«8
node tandem proiUt:
6)
1
1|
. {I— k-ibvni
l^L'tia'aml lim ami — i.^ ^ '\
\ i I V « ;
HiDC mutalo a in •»&, eniimiu:
.-.■..■.„( 4i).,...„(y )
nnde dlvisione facta:
t — k*niiMna.rinamH*MBunT. Mnam(u4'T— ■}
> . Moamu . rinamv . «inMnCa'f-v-|-*)
-^-'•■•"(^)-'-'(4^-') ■-^■-■-(4^)-^-"-(^ + -)
quae est transfonnatio {juaesite expressionis a CI. Legendre propoaitae in expressionem s). i
Formnlam 6), posito n, a, ▼ loco -^y^, -y— » ^~- — a, ita ijuoqiie reprae- '
aenlare licet:
8} 1— L*un>m(a+it). «'main(«— n) . «inam(a+t) . linaroCa— *) =
(t — k*«in*an*) fl — k*«iii'aina • lia'ami) '
tl— k'nn'snia , liB'ainut {1 — k'utt'ama ■ nn'am v]
-nnde formnla 4) nt cesos specialis fluit, posito a=:T.
by^oogle
IW
• ' 55.
E formulis $' aalecedeiitis \), i), 9)> 7) SHjuitur:
1) ntn, .)+n(v,«)- ncu+T, .) =
{'--"(^y ^-(4^ - •)} {.-v-:-..(4l)..i-^(^ ^ ■)}
{'— ■»(^)-"(^ + .)l {.-.....„(^).....,(^ - .)J
(1 — k'iiii'im(u+»).iin'«Bi[ir+«)]H — k*.i.'.m.«Q'»m(a + T-»)l ^
{1— k'ua'am(u_a).uD**iii(v— a))!!— k'Mii'«inaMn*un(n+T+>)} '^
— k'nnaiiiB . timmu . linam v ■ ■ipain{u-^ t — a)
14-k'iuiama . uDiiDu . naamv. niiain(u+T + *) '
-r'-n--
quod est theorema de Additiooe ArgumeBti AmpUtudiaU. Prorsiu eadem methodo inresti-
gari potest alterum de Additiooe Argameiiti Parametria at ope theorematis de reductione
Parauietri ad Amplitadinem, quod nobis rappeditavit fonnula 4) §. 52:
IV) n(u,a)-n(.. d) = uZ(.)_aZ{u).
e formula l) idem sponte floit. Etenim e IV. fit:
n{a, ») - n(u, •) = aZ(o> - I.Z{a>
n (b, u] — n (n, b) = b Z (■} — u Z (b)
n(a-H». o) - n(u, n+b) = (a+b)Z(o) - uZCapH-).
uode:
nCn. a) + n{ii. b) _ n(a, a+b) «.
nCa. u) + n(b. «) - n(a+b. u) + u(Z{a) + Z(b) - Z(.+b)t,
aive cum sit ex l):
__ _, . , I , 1 — k'*inainD.iiiiaina,itnaiDb.iinani(«-l-b'— w)
n(...)+n(b,.)-n(.+i...),-lo,. .^.^.^..^...^..^..^^......^iji.^.,; ■
porro e U.:
Z(a) + Z(b) _ Z(a+b) >= k>«inina . abaaMk. •{■•»(• +1>)>
fit:
S) n(u, a) + n(u, b)- nco.%+b) =
,, . , . , , ,, , 1- i_ 1 — k'«inainu . lin an a . md «n b . ■iiiain(a4-b— >}
k'unama>uiambnnaiii(a+b}.H + -r-tog • ■■ , ■ i . . : —: r— t V T-r-r-T •
ijuod est theorema quaeiiliUK de Addifioae Argumanti Pmmmetri.
Digitized by GoO^^jle
Alias eruimas formulas satis memorabiles consideratione seqaeute. Fit euim e
tfaeoremate III:
- ( ©CO) / " l_k'Mii'»m(u — a).«n'am{v — i)
I ec+'j-ecv+h) )*_ e(u+v+»+b) . e(a-T+.-h)
I ®(0) (" l-k'«n'«m(u+B).HD'amCv+b) "
lam e theoremate I erit:
n(»|-'. •+1') + n(d_.. .-1.) «. (.+,) z (»+.l) + (u_,) z (._b) + llo, . 8 (»+■—-■») ■ 9 (a-.->H.)
untie :
S) n(«+T, .H-bj + ncu-Y. .-k)-«n(.,i)-»n(T, l) =
(«+l)Z(. + b) + (._,)Z(,-b) - J„Z(.) _ «yZ<b) +
_i_ t-k-.f.-.»(.-.i..h-.„(,-b)
t ^' I_k'.r«-.m(u+.)..i.'.n.(,+b) •
sire cum sit:
Z(i) + Z(b) — Z(.+b) = k'.uinia.ns>i<.b.iiiinii(..|.bj
ZCm)— Z(b) — Z(i— b) = — k*>ui»iiii.iiD>mb.>iiiaia(s— b).
proJit 8), S):
4) n(n+Y, .+b) + n(— T. .-b) _ «n(.. .; _ in(,, b)' =
— k*Miiama . linambf dnam(a4-b) . (u+t) — ainam(a — b) . (a— y)}
1 . l-f».-a..(.-a)ri.-.»(.-b)
-r 1 -» l_k'.ip"an.(a + a).ia'a,n(, + b) '
Commutatis inter se a et t, obtluemus;
5) n(a+T, a+b) — n(a— V, a— b) — 8n(,, a) — «n(«. b)»=
— k>iiDama.iinamb{ainam(a-f-b) .(ii+*) + nnain[a— b) , (a— v)l
. J_io. 1— k*iin'am(y— a) ■ riii'aiii(a— b)
« ^' 1— k'ain'atn(r+a}.ah>*ain(aH-b} "
Addilis 4) et £) obtinemos:
6) n(a+T, a+b) — n(n, a) _ n(., b) — n(Y, a) — nfv, b) «=
— k'iina]nauaambaiaani(a-t-b].(n-f>T)'
. ' |„,f '-k''ij'a"("— a)'iii'ai»f»— b) 1— k'.iii'am(Y— a)al.'aii< (.— b| i
^ 4 ^>l-k-dr,'am(u+a)dn'am(,+k) ' l-k-d.-n>,(,^a) .i.-a«(.+b) j '
Digitized b, Google
• l-k'-rf
'«nl>
dn-
•■(«+•)
l-k*.ill'
ama
.!•■
«n(._b)
1— k'jhl'
'una
■'■'
.o<u+b)
l_k'«n'
amv
tin'
m(._.)
1 — k'un'
'■mv
lin'
».(.+•)
I-k»«n»i
unn
.jifl
•«.fv-.)
161
Posito T = 0, e 4), 6) prodit:
7) n(o, .+b) + n(...-b)->n(.,.) =
>uitin«nnuab|^am(iH-b} — ■iDain(a— b]]u •(
8) n(a, .+b) — n{ii, «— b) — sncn. b) =
_l'Ai«..am»nb|™™(.+b) + «.«.(.-b)|. + -l.|.,.
Posito b^o, e 4), 6) prodit;
9) n(.+T, .) + n(._T, .) - «n(a, .) = ikj.
10) n(.+T. •)- n(«— .,.) — fn(...)=.-i-'°e'
REOUCTIONES EXFRESSIONUU Z(in), ©(in) AD ARGUUENTDU
REALE. REDnCTIO GENERALIS TERTIAE SPECIEI INTEGHALIUM
ELLIFTICORUK, IH QUIBUS AHGUMEHIA ET AUFLITlrDINIS
ET PARAMETRI IMA61KARIA SUNT.
56.
Rerertimar ad Analysin fanctionain Z, @, qaarDm iiuigDem nsnm in theoria no-'
atra antecedentibos comprobarimus. Qaaeramos de TedactioBeex{ii:cs8ionDinZ(ia), 6 (in)
ad argnmentam reale. Idem primnin sigois O" Legendre nsitatis exequemur, deinde ad
notationes nostras accommodabimus.
Norimna in elementis §. 19> pag. S4, simal locam habere aequationes :
"» = ""«*• -jsr = !¥:?)• "*'- '"♦■■'■■'•
Hinc Bt:
nude iotegratioDe facta:
9 V*
/( r* k'k'fio'rl' 1
db, Google
163
sire:
I) E(y) = i(l6,;.A(.^. V) + FC./., k') - E(^!.. k^l.
MnItipJicomlo per ~^^ = -' ■ '''^^r et integrando eniimus :
Ex aeqaattone l) seqaitur:
F' E (a) — E* F f(cl _, f
'^ ^^L. ^ F'lg,(.A(^. k') - {f'ECv>. V) + {E«-F-)F{v.. k'>}-
lam adaoletur theorema egregiant Cl' Legendre (pag. 61 ): ■
F'E^CV) + F'(k')E' _ F'F-(k') = -^.
uude:
F'E(^, k') + (E--F')F{V., k") = -Zl_(F^(k')E(v., k') - E^(k')F(V-. kM + "^^^^^ ^'>
ideoque :
S) fgW-gTM = ^^,^1,, 1-, _ r-(t')E(v,,o_E-y)F(».ri _ .rty.- y;
'F' r'M tr'F'ft')
K Qotatione nostra .^rat :
9> = aman). V = •"n(it, k*), FCy) = iit, P(v-. k')eii;
poiTO:
F'E»)-E'F(y) __ F'MEftf.. l')-E'y)F(»., f)
F- ' ■ '• 775;; Z(".M.
unile aequatio 3) ita repraeseotatur ;
4) izf,«, V) _ _ig.„(.. ],■) a.„(.. k-) + jll, + z(„, n.
Hinc prodit integrando:
yij.ZC., V) = lo,„.™(u. fj + J^ + /'z(u, k-) J.,
db, Google
sire
cum
ait /"dnZ
w
= log
8(0) ■
"^
1-
= .
4KK'
nm{f.
.k')
»(",
«(»,
^
Fonnnlaei), 6) fanctionesZ(iu), ©(in) adargumenlum reale revocant.
57.
Matetnr in 6) n in u -H 2K', prodtt:
9f..+liK') JKK- , ... 9(11. k')' ' . K «M
sive posito u loco in:
Fooatnr in 5) n + K' loco n : cnra sit
kua*m(u, V)
ca*am(ii^-K, k) = i,oi(n, k')
prodit:
«('-+"')
JS /v , I., 90')
unde posito nnrms n loco in;
.(K'-»i.)
a) e(B+iK^ = ie /T.iii«m(u)eCa).
♦) Frt caim d(a+iK. k) = »Cii). idwxpie etlim eCu+«K. k') = «(«. k').
db, Google
164
Sumtu logarilhmia et dIfferentiaDdo, ex 1)) 2) prodit:
4) Z(. + iK') = ^=|^ + "!{!»". Wi"»W+ ZW-
Foiilo u =3 0, ex l) -4) 6t:
—
leciiK') = — e ^ e{0), e(iK') =
'' |z«iK')= ; Z(iK')= OO.
Formulae 1), 2) egregiam inrenioDt confirmatioDem e natura prodacti infiuiti} in quod
fuDCtioDem 6 erolvimaa:
\ . ; (i-i,c.ia.+,i(i-»q'ito.i.+,-)(i-»,-c»a.+,~) . . .
»ro l(l-q)(l-q'Kl-«.-.)" "
IB— l''"1(l-1'<''-)(l-q'''"') ..l((l-q»— '«)(l-q'«-")a-q'«-'"') .-I
ia-q)(l— l')(l-q') ..1* ■
Ubi enim mntatur x in x +— 7— > quo &cto abit e'" in qe'", abit productum
l(l_,,.ix)(l_,.,.r«)(,_,.,,ix) . ,.)j(i_,ir'i')(l-,>r.i«)(l-,'.— i«) ...) ■
in hoc:
.^^((•-i«'")(i-q'«'i')(i-q'f'») ■ ■ .)((i-q«-*'»)(i-q"«— ")<i-q'«-"») . • -I,
nnde:
Mutato rero x in x -^--j^t abit e'" in /q'e'*, nnde prodactum
((l-qe'"0(l-q'e'l'OB-q'.'l«)...||(l-qe— I'la-q-.— l«)(l-q'e--l«) ...|
in hoc:
(i-.-i«)l(i-i'.-i«)(«-q'«'") ...|((l-q'«--i')(i-q'.-'i')...l =
^.«.l..(t-Jq'coiti+q')(l-«,<o»t,+,-)(l_q-™t,+,") ....
db, Google
IN
At dedimiu §. S6 formulam :
■""■ « °° ^ • (l-t,co.J.+,'Kl-«q'»»t.+.l-)(l-!,'cai.J.+,»)--- '
UDde ridemas, fore:
Formulae 7), 8) aatem posito — — = a cam foimolu 1), 2) conTeniaut
E formula 9) §. es:
posito iu loco u, seqnitur:
e(!.+K) =
/f"o».l.(., k')
uude e 6) $■ 56 :
em /r »(». !■)
sive e formula allegata 9) $. 68 :
'' e(0) V 7 «(o, IT
Hiuc sumendo logarithmoa et differeutiando oblioemus:
10) lzr..+K)=^|i, + Z(.+K'.k').
■ «KK^
58.
Formularum §§. d6. 57 inTentarum fecilia fit applicatio ad Aualjrsio fnnctionum
n casibtu, qnibtts Argumenta sire Amplitudiuia sive Farametri sire ntrinaque imagi-
naria Mint.
Demonstremus primum, expressionem n(u, a-f-iK') rerocari posse ad n(a, a),
onde patet, posito n = — k^ sin' am a, integralia
db, Google
altenuu ab altero pendere ; qaod est iasigne theorems a CI. Legendre prolatum Cap. XV.
InTenimas : j
Fit autem el), 4) §. 67:
8(.— +iK')
©{.+ a + iK') «B.in(. + u) eC + u)
' nZ{a+iK') =! —Tk •" " Wg »">• -i*"" +«
ZC);
and*
J, terminis -^^ ^^ se destmeotibm:
1) n(a, .+iK') = n(u, .) + ucolg«iD«i«o..+ i-[(.g
■in air
'(»-»)
'(«+•»,
Fooamns in hac formula ia loco a, fit:
liLiunri.linmfial .- -i^'w-C". k")
t«ig«i.[i.)A»m(..) = ^^^^^^^ k')co.a«{., k-)
«n»in(i» — 0) i»iiiu — cotgam(i«)Aimfia;tgamu
diiam(ia + i') Aamu + cotg am (i>) ^unCia) tgamu
sire
posito breritatis gratia:
^•>n(-. kl . -„
•inamC., k')cM.m{.. k')
fit:
»m»in (■•—") iimu + i/"«tg»mu
Mi,.ni{i. + ») A«mQ-iv^«l«amu '
nnde l) abit in
n(u.u + iK;)-nc„.U) ^,^_ ^ ^^
/••"«'
una
quae cum formnla f) a CI. Legeodre exhibita coDTenit.
d by Google
■ 59.
Alias formulas , pro reductioae Argumeuti imaginarii ad reale faodameDtalea , ob-
tinenma e 9), 10) §. 57. Qaarum primum observo hauc, qua ArgnmeiLta et Amplitudinu
et Farametri imaginana ad Argumenta reatia revocautor:
I) n(i»,u+K) = n(n, .+K'. k-).
quae hunc in modum demonstratnr. Fit enim:
nr,.. i.+>c, = i.z(i.+K, ; ±..,-111^1^,
porro e lo) §.67:
i.zr,.+i) = ^^ + .2(.+r. k').
e 9) §.«r:
_ .(.—>)■
9(1. -U+K) fV «K' eft-.+K'. >•)
V I' ' »Ul.l')
'(•+'■)•
«KK' 8(i+u+K', V)
9(0,
,k)
9(i.+i
.+K)
9(0,
I)
80. -i
• + K)
YF-
_ KE' e(»— M+K'. k')
ideoqne^ termisis . — ' ^ , - se destraentibiw,
nc.«. :.+K) = nZ(.+K', k') + -i-iog ajI+r+K\kO " "'"' '"*"^'' ^''■
quod demODSlrandam erat.
Mntato in l) a in — ia, prodit:
^ nc»ii.i+K) = -n(o,i«+K', k').
Fonnnia l) facile etiam probatur consideratioue ipsins integralis , per quod
functionem 11 definivimns:
nc..),. /"""-— ;-^--;-"-""'-'" .
^/ 1 — k'nn ■■!■ . un'ama
Digitized by Google
miu, i«^ ; J 1— li'Mn'.inp.+K).«D'wii(iB)
O
Fit enim e fomnilis §' 19:
wnun a«+K) = •ineoam(ii) =
Aw.(«+K'. k-)
e«»mO«+K) = -coJM«io{i») = — j|— cwco.m(«. k") = -^co*«in(«+K', k'>
A am r>«+K) = acoain(i>) = k'>tnca>in(a, k') =■ b'sinain(a+K', k*).
node:
ikkn>am(ia + K)co*amQa + K}Aain(ia + K) ai
— k'k'MnamCa+K', k')c<>i»in(a+K'.k')Aam[i+K',k').
Porro fit:
"°'»mCio) ^ -t g'amdi. k')
1— k'Nn'amfia+K) riD'amCin} l+&*am(a+K', k'; te'aiii(u, k'j
-ri.i'an.(n. k') _,ii.'.m(i., k')
co.'am(ii, k') + a"am{a+K'. k") i(o*a«.(B, k') 1 — k'k'«n'ain (•+K'. k') ijn'an (u, k') '
nude:
nCin.ia+K) =
/■
k'k'»iiiini(a+K', k').co»ain(a+K'. k') . Aam(«+K', k') ■ HD'am(ii, k'j.du
l-k'k'HO>an,(a+K'. k") .m'am(u. k')
Sire:
n(iu, i.+K) = nCtt. a+K'. k').
quod demoDstrandnm erat.
E formnlis 9)^ lO) §. 57 simili modo atque l) comprobare possumus formtJam se-
quentem, quae docet, fuactionea bioas ArgnmeDti imaginarii Farametri, qnanun Moduli
alter alterius ComplemeDtum, ad se ioTicem reTocari posse;
s) incu. ia+K) + in(a. io+K'. k') =
^^ + »ZCa+K', k') + aZ(u+K. k).
Fit enim:
inc. iM-K'. ll = i.zr.u+K'. 11 + i- ,., . |g:+g-';g ..
,, Google
169
lam Bt:
k 4KK' »(.+iii+ lf, IT)
k'" — e^ri^ —
9(i.-t-K+ii) 9iii.-ii.).|-K| /i 4KK' 9(._i.+lC, !•)
em «m ~Vk'' w^ii — ■
unde cmn sit e(m-K) = e(K — u) :
e(i.+K+.) ' er,.+K'_.,k') ■
9(;.+K-«) , ■ eC"+K-,,V)
iileoque;
T'°«- 90.+K+.) ■t-T"*- 9r..+K'+.,k-j = ~ siii?-
PoiTO fit:
i.Z(i.+K) = Jl^ + .ZJ.+Ck')
i i/Z(i«+ICk-| = Jl^+.Z(.+K,k).
node:
int.. I.+K) + inf., U+If. k') = ~, + "ZL+K', k') + .Z(u+K. k)j
tj. d. e.
60.
Patet e formulis:
Mnam(K+iu} = — aeoaro(a, V)
«,«,(.+iK') = i..-j±_,
Argmnentnm u, quod, dmn sin am u a jmjm ad 1 crescit, a ad K tranait, uU
sin am a a 1 naqna ad -j- creacere prgat , imagiaarimn induere valorem formee
K +iT, ita ut simnl y a naque ad K' crescat; deinde crescente sin am u a -i-
T
DigfeedoyGoOl^jle
170
tuqae ad CO, induefe a formam v-f-iK', ita at simol t a K tuque ad decre-
Kcat ♦).
Hiuc videmiu, siquidem in tertia specie Integraliam Elliplicorum, quae sche-
male contenta est:
/i
(i + 11 tin' ip) A [qr) '
ponatur, uli fecimus, u = — k'sin^ama, quoties sit d Degatirani
inter et -> kk, poDi debere a := — k'sia'ama
- —kket — 1 , - - n = — k'sin'aei(ia-i-K)
J et — 00, - - n = — k'siD'am(a-f-iK'),
desigDauIe a qaaDtitatem realem. Porro cam sit — kksin* am(ia) = kk tg* am (a, k'),
patet, quoties sit o positivum qaodlibet, poni debere:
a ^ — kk siD*ain{ia).
Hinc quatuor classes lotegraliam EUipticorom tertiae specie! nacti sumos, qaae respondeat
schematis, qnae Argumeata indaant
1) a, t) i. + K, 8) • + iK'. 4) i..
(|aarum tres primae pertiaeot ad n negativum , quarta ad positirum.
At per formulam l) §. 68 videmus, fanctiooem n (u, a + iK.') redaci ad n(n, a},
sire classem tertiam , in qua n est inter — 1 et — 00 , reduci ad primam , in qoa n est
inter et — kk. Porro e formula 11) §. 6S **), funclionem n(u, ia) semper reduci
*) Oktinekilur rimnl:
!„.„„. 0. ' . 1, ' !_. 4./!+?. »
.-0, Ji- . K. K + i^. K+iK', -5- + i'''- '»'•
**) Haec formula icUic«t, poiito ia l»co a id tcqucnlem abil
n(u.i.+K)-n(., i.)
^ — > + Arc Ig {« iiD am u . an coam u| >
nC'.k*)
- — 'l ■ . Quae facile per fbmmlai eleneiitare* {' 19 luccedil Iniufor'
db, Google
171
ad n(u, ia-hK), sire clasiiem qnarlam, in qua n est positimm ad itecuudain , iu qna
n est negativnm intra — kk et — i. Unde iam narti samas theorema, propoaitum
ittttgraU
f
ti+ii»i»'»ja(v) '
quaecunque tit n quantitat rttdia poaitiva tea negativa, semper redact poaae ad integrak »i-
miU, in quo a negativum eat inter et — 1. Qaod e&t egreginm inventum Cl' LegeDdre.
Iam Tero consideremiu casnm generelem , qno et AmpHtado et Parameter fbmiain
habent imaginariam qnamlibet: comtat, earn casum amplecti expressionem
n(a+i». ,+ib),
designantibus n^ v, a^ b quantitates reales. At e formutis §' 66 videmas, eiasmmli ex-
presuonem redaci ad qnataor hasce:
I) n{a. .), t) n{iT. ib), S) ncu, ib). 4) n(iY. «>,
rel, si placet, ad qnataor hasce:
1) n(u, •—10, ») nciT. ib+K)
8) n(it, ib+K), 4) ncit. a— K).
G«aeraliter enimexpressio n(aH-T» a+b) ruexpressiones n(a, a), n(T, b), n(n, b),
n(vf a) redit, e qnibns qaatnor propositae prodeunt, siqnidem loco r penis ir, locoa, b
TCTO a — K et K+ib. Poiro e formnlis 1), 2) §' 69 fit:
npT. ib+K) = n{v. b+K'. V)
n(iy. .-K) = -n{», i.+K'. k'},
nnde expressiooes l), 2) classem primam redennt n(u, a), expressiones S), 4) in clas-
sem secnndam n(u, ia + K); id qaod nobis suppeditat
THEOREMA.
Integrate propositum formae
r '■f
quodcunque sit n et (^j aive reale sive imaginarium ^ revocari potest ad in-
tegraXia similia, in quibus et (p reale et n reale negativum inter et ■ — 1.
Y 2
Digitized b, Google
1T« ^
£/hoc dworema debetnr Gl" Legendre, iiisi quod ille reales tantum Aaipliludioes
coutemptatus sit.
ITormuIis 4), 5) §. 66 redacitur n(uH-v, aH-l>)-|-n(u — t, a — b) ad n(a,»)
et n(v, b), n(u-4-T, a-t-b) — n(u — v, a — b) ad n(u, b) et n(v, a). Hiac pa-
let, posito
n(u + iira+ib)+ nc«-iT,i-ib) = L
n(u+ir.a+ib) - n(u-iv, a-ib) _ ^
pendere L a funclionibus n(u, a — K), n(iv, ib + K), Ma fuoctiombas n(u, ib+K)>
n(iT, a — K), ideoque redire L io classem primam, M id classem secandam.
Haec sDDt fundameota tfaeoriae tertiae specie! lutegralium EUipticomiUj e princi-
piis BOTM deduct!. Alia infra videbaulur.
FUNCTIONES ELLIPTICAE SUNT FUNCTIONES FRACTAE.
DE FUNCTIOKIBUS B, S, QVAE NUMERATORIS ET DENOMINATORIS
LOCUM TENEMT.
&1.
Evolationes §. S5 exhibitae genaiDam functionDin EUiptiearam natarani declanmf,
videlicet esse eas foDctiones fractas, ut quas iam ex. elementis nOTimiis, pro innumeris Ar-
gnmenti raloribos inter se direriiis et eranescere et ia infiDitum abire. Iam antecedentibus
ad functionein delati sumas , quae fractionis , in quam evolnmus ipsom sin am =
^ ' (1— «qeo.«i + q")(l — «q'c<Mt«+q")(l — tq'cM«K + q") ...
deoominatorem constituit, fuQCtionem dico
>m
g— tqc<Mt» + q')(l— tq»cotti4.q«)(t — gq'coi<i + q") .
eCQ) (a-<()Ci-q')Ci-q*)a-q') ■ ■ -P
Iam et aameratorem particulaii charaetere denotemns, atqne pooamus:
' « ) _ « yV«mx(l— gq'coit» + q*)(l— 8q<co.«i+q')(l— Iq'cwgt + q") . ■ ■
«{0) «i-q){i-q')(i-V)a-q') ■ ."T? *
Digitized by Google
ReJitjau advocatis erolutionibas §. 85 traditis, inreuimusr
/v.
node posito = d:
Hinc flnuDt formulae speciares:
t> »(K) = -2^1 HW
Yf«
Fosito H'(u) = ■ , cam sit;
ffCu) =/Vco..m«a.mn e(u) + /Tdnamu e».
pro valorihtu u=0} a:=K obtiDemua:
E s) seqnitar adhoc:
Cetenun fit:
6) H(.+«K) = H(-u) = -E(u)| H(.+4K)=HW.
*) Fit aim Z(K) = 0, uiHh etwn tf<IQ = 6(K)Z(KJc=0.
, Google
1T4 —
E formnla !) §. 57:
»(g-«iu)
9{u + iK') = i. *" /^.illM.D^ii.1
seqnitur:
7) e(.+iK') = i. HW.
Motato IQ hac fonnnla a in u+iK', et adrocata 1) §. 67:
«) 9(a+SlK') = _e " »(.).
prodit :
S) H(u+iK') = i. *" e(u).
ande mxsiu matato n in u+iK', e 7):
,(K-i«)
10)
H(.+JiK') = -
., " H,«).
! fonmilis 7) - 10)
derivari possnnt generaliores :
11)
irun
ill')
le)
.«"■.„ = ,.
.(.+(«n, + l)iK')'
,iK)
ij)
. ""■«<.) = <-
.r(u + (S„+l)iX')-
9(.+(I.
■+l)iK)
U)
.""■«<.) = (.
_,)..*.. «■'■
«(.+(« ii, + l)ill').
; 12), IS) «l:
15)
»((«m+l)ilC)
= 0; H(SiiitK') = 0.
Formalae 5), 6) demoQstrant, faDctiones 6(u), H(n) tnutato u in u + 4K
formulae ll), 12), functionea
db, Google
176
mntato n in n + ^iK'iniroutatas nuDere; ande illae cam ionctioDihaa Ellipticis alteram Pe-
riodam realem, hae altenun Periodum imagmariam communem habent.
E formula 5) §. 66:
8(1.. k) 4KK-
s(o,k) ' "•■"
<-'>^|7^-
seqnitar:
Rum
-|5lL^-/Twn»r...k,.
4^ = ,.^/T..„<..k-,
9(1.. k-)
■ 9(0. kV
unde e l):
irua
,_ 95.. I) ,Ak 4KK'
"l -9^75" - V f
B(.+K'. k-)
9(0. k')
m '!:■'} =i/,^.^.
H(«, k-)
6(0, k'j
E 16) sequitar, mntato n id ia, et commntatis k et k':
' eco. k) V P" ' e (0. k*) '
cui adiangator 9) §. 67:
.« gPo+JC.k) _,/r "itF «Ca+K'. k-)
' »{o. k) V k' e(ork^ ■
E formula supra iuTeuta:
seqoitar:
Qua ducta formula in
k«i»ain(ii + *)Mawn(u — t) = -
1 — k'tin'amu uD'amv
prodit :
81) H{ii+,)HCti-*)
H-
new-
. e-E
iH'»
«>
»»«, -
- H*«
.HW
H'«®'
> —
e'DiPT
db, Google
DE EVOLlfTIONE FDNCTIONDM H, IN SERIES. EVOLUTIO
TERTIA FUNCTIOHUM ELLIPTIC ARUB.
62.
EvolTamas fonctiouev
*(ir) (l-gq(».». + ,^(l-t^'co.e. + T)(l-»q-.c^«. + ,") ...
em t(i-q)a-v)(i-«...r
["^j »'/Vlilli(l-gq'co.«.+q')a-«<l'c»-«i + il')(l-»S'c.oi«i + q") ...
em l("-q)(i-q")(i-i')...)"
(— )
0(0)
e{0)
- = «V^f'*''"»'' — ^«'>'" + ^«"5' — "*«"'■* H 1-
DetemuDatioQem ipsarum A, A', A", A'", ..; B', B", B"', B'*, ., tianciscimur ope
— irK'
aeqaationmu 7)- lO) §' antecedents, quae posito u= ~-^, q ^ e in m(|uea-
tes abennt:
.(i£i) = _q,..«»(i5L+.,K.)
Quam in finem erolatioiies propositas ita exhibemns:
= A - A'.'l" + A-e"i« - A-.'l- + A~."" - . . .
_ A'.— 1« + A'f-I" - A-e-'l" + A-e-l"
, Google
MaUto X in x — ilogc), aliile"'" in q"!!""", e'"" in *~"" ; porroe(-?^), ul'"' \
in © (— ^ -I- 2 i K'j , H(— j^ -I- 2iK'j. Hinc nanciscijnnr:
9(0)
'
B[0,
-*,.•!«
+ A',
>««))■
— A'.
!'«'
.U + A-q
.,.i«-
.-.-,.
-f
t-*iK
•-
■'.^.
.-.!._
8(0)
il^
— qe
• ix. _
!l:
9(0)
1 =
•/v
!-"-
ff,'e»
" +
B-,.,
llK
_B-,.,'
"■+ •■
•f
vi-
{?--
1*
*" +
B~ _
fix
B"
"+■■
•}■
Qiiibns cam expressionibiu propositis comparatts, eruimus:
A' = Aq, A'ssA'q*. A" = A'q«, A" = A'q'
B-.sB'q'. B"=B-q«, B"=Brq-, IT" s= B"q«
icieoqae
A' = Aq , A" ^ Aq*, A™ = Aq*. A"" ^ Aq", ....
B"= ffq', B"'=BV, B"" = B'q". B"*=!B'q'".
ande evoluliooea qnaesitae fiuot:
0)
= Afl— ZqwZi + 3q*cos4i — 2q*cot6i + fq^cMSi — . . .{
: Syq" Jff""" — q'na9» + q'-'sinSi — q>-*sin7i + q«-^*in9i — - .J
EvolQlioaeA inventas alteram ex altera derirare licuisset ope formnlae:
db, Google
tT8
mutfiudo X io X — ilog/Tf, quo facta e*""*, e •"'* abeunl in q^e*""*, - — 5; — , ©| — ^j
ui6[— ^ — h-iK'l, et nraltipUcatido per vq e'", obtinemua:
9(0) * ''* ©(0)
q
8(0)
(«V^*wii — sV^uaii + fV^fiiiSi —lV^fna7t + . ..J.
9(0)
Qua insuper Aualysi eraimns:
63.
Determinatio ipstas A artificia particalaria poscit. Fooamos, qtiod ex anlece-
dentibns licet:
(1 — «qCM«« + q^(l— tS'C0.1<-(-q»)(l— «q»eo««i + q'») . . . e=
P(q}fl_2qeoilx + Sq*ca«4i — Zq*OM6i-^Zq'*CM6i4-. ..| .
ibi(l— Sq*c«j£t-t-q*}(t— Sq*CMfx + q')(l — Sq*coiti + q») .. . =
P(q) {mdi — q'-*»mS»+ q» •»iin5i — q»-*Mn7< + q«*'«ro9x —...);
fit:
P(q)
ta-qja-q-Xi-q')."**
Expressio secnnda immatata inaaet, ubi ducitar in primam, et post £Ktum productom po-
nitnr q' loco q. Hiac obtinemus aeqnatiooem identicam :
P{q')P(q*)("n« — q*»i'»3» + q'*««'5i — q»«iui7< + ...j x
{1 — Zq°c«*2i + Sq*cof4i— {q**co«6x4-. . ,J 53=
P(q)(«nx — q'»mSx + q*iiii5« — q"»iii7i + . . .(.
dbyGoo^^jle
IfQ
Ipsam iam iostitoamas nmttipIicattOQem , ita nt uLitjoe loco 2 sin . mx cob . nx sen-
batar sui(mH-n)x + sin (m — u)x: facile patet, Coefficientem ipsins sin x io pro-
ducto eroluto fore:
i + q' + <i' + q"+«r + -...
ita ut prodeat:
At iDTenimua e secnnda fonDalannn propositanun, posito x = ~-:
{{i+q')a+q')a+'rt ■ . T = P(<i)(i +q' + q* + q" + q*+ . . .J.
unde :
^;^=„.+fl(.+,-)(.+^,...r
sive:
IM. = (l + q-Jd + l-Kl + q-) ...
(1^,0(1_,.) (!_,•)...■
Hinc e methodo iam saepiiu adhibits *) seqnitnr;
<i-q')(l-l'l(l-'nci-l')...
Hinc tandem prorenit;
a-q')(i-.l')(l-<l') . . . ((t-<i)(t-i')("-q') ■ ■ .1"
(l+1)('+q')('+'l')('+l')... .
d-^tt-q-Xl-q-Xl-q') ... '
sive ex iis, quas §. 86 dedimus, erolationibna;
1 /tTK"
Qnantitatem illam , qaam faactenus indetenninatam retitjmmua , 6(0) porta
i:
znus iam:
*) Viddicel ponendo *Dcce*Mve tf, q*, q*> q'* • . loco s ct inttiiucDdo malltplicjlioncm inrmitam.
z i
, Google
180
64.
Aequationem iileoticam, quani antecedentihus comprobatum irimus:
(l-!q«o.!.+ ,')(l-!q'«,.J. + ,-)(l-«q'cc..li+,») .... =
1 — gq'co»8» + »q*CO><» — »q'COs6l + gq"cw8l — . . .
(l_q') (!_,•) (l_,')(l-q') ....
alia adhuc via, a praecedente omaiDo diversa, jnvestigare placet. Quam ia fiaem
tamqaam lemmata antemittamus formulas duas seqnentes:
1) (i+q')(I+q'")(l+q'»)(i+q'-) ■■. =
1 +
l-q- ^ (l-q-jH-q-) ^ (l_,>) (l-qVd-q-) ^ (l-q'Jd-,-) (!_,•) (l_q-)
(l-q.)(l_q>.)(l_q>.)(l_q'.) .
q*
q 1-q. ^ (1_,)(1_,') (l_q.)(l-q'.)
J! •'. J. .
+
(l_,)(l-q1(l_,') ■ (l_q,)(l_q'.)(l-,'.)
Ad demonstratioDem prions observo, expressioaem
(l+q-)(i+q'-)(l+q'-)(i+q'") . ■ ■
posito n' z loco z et muItiplicatioDe facta per(l+qz), immulatam maoere ; unde po.sito ;
a+q')(l+q'«)(i+q'') . . . = l +A'.+ A".' + A™.> + . . .
eniitnr:
l + A'. + A-.' + A-.'H....=(l + ,.)(l + A-q'. + A"q>.' + A"q>.'+...),
ideoque, facta evolutione;
A' = q +,q'A', A" ^ q'A' + q*A'', A"' ^ q'A" + q"A"'
sive ;
l-q"
. A"
db, Google
1-q' ■ (i_,')(l_q«J ■ (I— l')Cl-V)a-q*) '
niculi propoiiitum est.
Ad demoDstratioDem formulae 2) observe, expressiooem
(l-q.)a_,"=)Cl_q>t)Cl-q*l) .... '
posito <j z loco z et muldplicatiooe facta per -r— — , immntataoi maoere ; unde posito :
' + -i-::r7+-
(l_,.)(l-,-)(l-q'-H>-q'") ■
A"«>
l_q. ^ (l_,.)(l_q'.) ^ (i-q.)(l-q-.)(l-q".)
obtiDemua :
^ l_,. ^ (l_,l)(l_,'.) ^ (l_q.)(l-,'.)a_,..) ^ • • ■
' A-q. *"q''' J A"'q'.'
i-qi ^ (l_q.)(l_,'.) ^ (l-,l)(l_,-.)(l_,..) ^ (!_,,) (l-q'.jd-q'.jd-q-.l
_ 1 J. (q+*'q)' . (q'A'+q'A-).- (q-A-H-q-A-).'
■^ 1-q. ^ a-,.)(l-q'.) ■*■ (l_,.)a-q>.)(l-q-.) ■*■ ' ' ' ''
Hiuc Suit:
A' = q + A'q, A" = q'A' + q'A", A'" = q' A" + q' A'"
ideoqne :
A = —3- A- = '''*' A™ = ''*"
1— q ■ 1— q' ' 1 — q' ' •* '■
tinde:
A- = T^!-, A" =
1-q ' fl-q)(l-ql ' d-qXl-q-Xl-q") ' ' "
sicuti propositum est.
db, Google
lam fonnemus productum :
ll J 5 . E 4- 2 ^ — . t' 4- 3 . s> -1- . . .^ y
I ^1-,' ^ (!_,•) (1-q-) ^ (!-,■) a-,') (1_,-) ^ )*
fi + -a-.-!- + - l'^.i- + £ ■- + ■■■]■
Coefficientem ipsios z" sive eriam -j-, qnem ponemus B'"', eruinias 8e<|aeutem : B'"' ;
(!_,■) (I-,.)... (! — ,■")
r^T^ +
q" i'
I (l-q-jd-q-Kl-q.) ' (l_,."*')(l-q"*')(l -q"") ^ ' 'I
At e fonnnla 2), posito q^ loco q et z = {{'^, expres«ioQem, quae uDcis incliua coDspi-
citor, iQTenimas =:
1
(l_,.n..)(l_,.".)(l-.,.»*.)(l-q"*-)... ■
node
ideoque :
{(l+q.)(l+,'.)(l+,'.) . .j {(l + -2-)(. + -S.)(l + Jl) . , .} =
. + qf+-|-) + q-(-+4.) + q-(- + -^)+...
(l-ql(l_q'Kl-q-)(l-q')...
sire posito z^e*'", et mutato q id — q:
(1 — IqColJl + q")(I-!q'CM!. + q')(l_J,lco.!. + q") ... =
1 — gqw*» + gq*C0t4» — tq*cos6. + ■ . .
(l_q')(l_,.)(l_,.)(l-q') ...
Quod demODStrandom erat.
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iSS
Ubi ponitur — qz' loco z atque per vq 2 multiplicatur , prodit;
VT(" - -f) {» - 1"'")!' - '!'■■)<' - I'-l ■■■]"
yV(.-4.)_y?(..-.L)^.yv(.._±)_....
(l-ql(l-q«)Cl-q^(l-q*)....
sive posito z^e'*:
£/q """(I — *l°««ix-|.,q*)(l — Sq*G(MSi-t-<|'}(l — Xq*c<MSi-f-q") . . . ="
d-q^Cl-q'Xl-q'jCl-q')...
(juae est altera erolutio inventa.
65.
Evolutiones functiooum
1) e|^-^^] = 1 — Z^CDiSi + Sq«co(4i — Xq>oM6> + 2q"co*8i
2) h/. ~\ = i^/~qnnx — tV*?™'' + sV^woS" — sV^""?" + ■ ■ ■
spoute ad eTolotioDeni nOTam fuuctionuni Ellipticarum dacuDt. Eteuim e formulis l) §. 61,
poneodo a =: ~ , obtiDemns :
. ^ . '{—)
db,Goo';5le
ande:
« /T 1 — 2qco,«i + 2q*co.4. — tq«co»6i + «q>" w.8t — .. .'
41 co.«ni3lL = ,/L a*^" " + g/V co»S, +g*/^co,6, + g/^co»7, + . .
« V k ' 1 — 2qco»2j< +24'co.4» — Sq-'coifi. + iq-'cosS. — .. ~
51 A ■«> H^ = yV 1 + gqco.ax + gq'eo.4x + aq'co.6i + gq»CM8^ + . . .
' T ^ ' l-8qcoi8i + «q'co.4i — Sq-coiex + Sq^coiSi
Porro e 2), s) §. 6t , cam positiim sit ©(O) =y , obtiuemus:
«(K, = /f, H« = /!p, e(o, = /!p. .-TO^/niE.
unde e 1), 2):
8) y ^ = 1 - S, + Sq' - Jq- + «q» - !q" + . . .
"J y ' '■ ('T')' = '"^ -«'/? + 'oV?! -"*/?: + «■/?:-...•),
unde etiam:
l + !q + Jq' + !,' + J,"+ ...
11) /y = ' - '1 + '1' - »q' + '1" - 'q'' + ■ ■
' + 'q + »q' + Sq- + Sq" + Sq" + . .
Fit porro, cum sit Z(u) = ^^, n(n, a) = u Z (a) -t- -i- log ^'°~'' ,
Tt \ It } 1 — SqcosSi + 3q*coi4i _ Sq<cos6x + Sq>'>coi8T _ . T
•j EtCDim cum iil — s= .
dbyGoo^^jle
l_ 1 -tqco.t(,-A).Hq*eoi4(— A)-tq«co«6(.-A) + .
« ^ 1 — tqcosf (k+A) + tq* «»4(x+A) — Jq- co.6(. + A) + .
Qaae eat erolutio tertia fanctionam Ellipticaram.
66.
£x eTolationibiu ioTentis;
1) m-<f>ft-<fXt—'f) -.lid— «?"»•■+ q')(i—«q'coili + q-Ki_t,'c<»s. + ,") .,.)
1 — tqcuti .f- Cq*eM4i — t<|*cu6i + Sq"co«8i — . .
til— •fin— ^11— <f) ..)im>a— i<|'cciti + q*Xl — Cq-coiIi^-q-J .. .
(ini — q*tiitSi .f> q* MnSs — q" »iii7« + q* iiii9» — ...
quarum postremam, posito /q* loco q, ita quoqae exhibere lic«l:
1 1(1— q)(l— q1(l— q') . .l«iu(l — •qcmt. + qld — tq"«o.J« + <i')(l _tq'c<.it, + ,-) ...
■iDi — qfinSx -(• q'rioSl — q*iiil7x .^ q*" nii9l — q" tin Hi 4- ... ,
sequitor, posito x = 0, x:
I - 8q + f q- - tq- + tq»
(i-q)(i-q')(i-q')(i-q')
(•+q)(l+q*Kl+q')(l+q")
(l-q-jd-q-Xl-qlP-q') ■
d-qXl-q-Xl-q-Jd-q')
5) ld-q)d-qOd-q')d-q')-
1 + q + q' + q" + q" + q" + • . .
= 1 - 9, + s," - rq> + 9," _ . . .
Pouamiu in 2) x^~-, fit »inx = + V -j-, sm8x = 0, siD5x=— y — ,
= -i- y -J", cet. ; porro (1 — q)(l — 2qcos2x-4-q*) ^ t — q', uode 2) id hauc alnt
fomiolatn :
d-q-jd-q-jd-q^d-q") . . . = l-q'-q* + q'' + q"-q"
sive:
6) d-q)d-q^d-q-)d-q')- . . - l -q-q^+q' + q' -q"
A a
,, Google
ouiiu sei'ivi tcrmiaus generalu est:
(-l)"q *
Comparatis iuter ue 6), 6) obtinemua:
»)(l-q-"l' + q' + q'-q" f = l- St + i<f -7<f +3<f - . . . .
Formulam 4) etiam CI. Gauss iuvenit iu Commentatione : Summatio ' Serierwn
quarundam aingtdarium. Comm. Gott. Vol. 1. a. 1808 — 1811. Quam ille deduxit e se-
qaeate formula memorabili:
8)
(l-q.)(l-,'l)(l-q-l)(l-q'.) .
(l-«(l-q-)(l -,')(!-,')...
q(l— ) . q'(l-.)(l-q.) q-a-lXl-q.Xl-q'.)
^ 1-q ^ (l_,)(l-,') ^ (l_q)(l_,.)(l_,.) T •■
posilo z =(j. Coi addi {lossuiit formulae similes, qoarum demoustratiotiem hoc loco
omitto :
„ » (l+l)(l + q.)(l+q-.).. 1 a-.)(l-q.)(l-q'l)..
« (l+qJd+q-jU+q').. "^ « (1+q) (1+q*) B+q') . .
_ q ('-■•) . q<(l-.')(l-q-.-) ^■(l-.-)(l-q't1(l-q-.-)
l-q- ■^. (l-q-Jd-,.) - (l_,')(l_,.)(l_q.) "^ ' ' '
,(. _3_ (l + 0(l + q.)(l + q'.).. q (l-l) (1-qi) (1-q-.) . .
I< • (l + q)a+q'Hl + q').. " Si ' (1 + q) (1 + ,1 » + q') . . "
q'C— •) . q-H-Od-q'.-) l"(l-.1(l-q-.-](l-q..-) ,
1-q- "^ (l-q-Xl-q') ~ (l-q1(l_,.)(l_q.) +■•
qaamm 9) , posito z = q , praebet :
1 _1_ (1— q)(l-,')ri-q') ■ ■
I ^ « • U+q)(l+q')U + q>).
(l-,)(l-q-)(l-,.)(l-q-) .
(i+qXi+q'Xi+q-Xi+q*) ■
:l_, + q._,.+ .
l_I, + fq<_J,' + ,
quae est formula s).
Formula 6), quae profundissimae iudaginis est, ut quae a trisectioae fo&etionum
Kllipliuarum peudct, iam e loogo tempore a CI. EuUr inreota est ct luculeoter demoo-
.slrata. De qua iusigux demouslratione alibi nobis fusius ageuduni eriN
db,Goo';5le
187
His addamus evolutiones sequeutes:
V ^^'( . ) «yq'l('-i')(i-q')tf-i')(i-q') ■I'
eCi£i\ " !i-«i"»>-+q")('-«q''»>«-+q')('-«q'"-*-+q") ■
»'/?a-q') «W(i-q-) «y?(i-q'')
1— 3qCMtl + q> ~ I — fq'0O»ll + q« ''" 1 — tq'colSl + q"
{(t-q')Ci-q«)(i-q^(i-q') •
1(1 — tq'coi«i+q«)(I— »q*CMSx + q"){l — aij«co»a. + q").
_i «q*a+q')»i» ■ 4q'(i+q*)»i..x *s"a+q')«i'" .
lina 1 — Sq*cci*Sx + q* 1 — S^^cmSi -(- q* 1 — Sq*Mi»l» + q*'
- _ 1 r (l-q')a-q*l q'(l-q«)(l-q') q" (1 ^ q') (1 - q") _ 1
" HD* \ 1 — 2q'Goati + q* 1 — 2q*coj2x -h q* 1 — tq"eo»*i + q" ** ■ ■ j-
quae e nota theoria resolatioois finctioiiam compositanua in simpUces fecile obtioentur.
Hinc dedacoDtnr eTolationes speoiales:
l+q ^ 1+q' 1 + q' "^ l + q«
Quilius cum erolatioDibiis expressionum , supra exhilutis comparatis , prodit :
V? vq^ . V q' /V
l-q 1-q'
+
_q. l-q' ^
^(S)-^(4±f)+^(4^)--
., 4,. 4,. 4,.
i+q ^ l+q- 1 + ,' ' !+,■
4q 4,. 4,- 4,..
I+q ' l+q" l+q' i+q-
Simili modo CI. Ctauatn nuper oliserravit *) ,
seriem
/ + -J* + ^' + /* +..,
*) CreBe Jonmil eel. Tom. III. pag. 95.
An 2
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188
traostbrmari posse in hanc:
InTenimus snpra erolntiooes ipsomm , — eorumque digmlatum secuiidae,
t«rtiae, quartae in series. Quae igitur evolutiones dignitatis secundae, qiiarlae, sex-
tae, oclavae expraasionam
V 'T~ == » + «i + *>!' + «i' + «q" + ■ •
V^"^ = «V"q + * V? + t'/i^ + «V^ + . .
suppeditant,' unde vaiia theoreniata Arithmetica flauot. Ita e. g. e formula :
(■?')'= {* + *•' + *■)* + *■>' + *1" + • •}' =
uht p uumerus impar cjuilibet, (P(p) sunima factorum ipsius p, fiuit tamquam Corol-
larium tbenrenta inclytani Fermatianum , nUDienim unnmquem(jue esse sunimam qua-
tuor qiiadratorDtn.
TVns ExrBES5«M OF.BAlIt:RII.S.
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IBS
CORRIGENDA.
Lectorem benevoluin orntum rolo , ut ante Imtloiiefn corrignt , quae Irrepferimt menilB gravlora
seqneolia :
P>g. 3. lin. II. leg. M loco U, bi*.
— '■ - '•"■'»B-(I+^"«-C-(IT^-
— 8, loco k' leg. i'.
— 8. — 6. loeo /"kT7 leg. /^.i, bis.
— 9. U et V ubiqae inler *« commulari dcb«nl.
— 10. — S. *. 6. loco — /V leg. + /T-
— 17. — 8. loco »"y, l>"j leg. ""y'l Vj\
-«5. _ 17. loc(.(t+2«)'l.g. {I+S«)'.
- f9. — 5. loco + 1* leg. — T*.
— 7. loco: V loeo u, leg.: u loeo v.
— 18. loco a — u'v*J leg. (I — iu'v").
lia. poitr. loco:
,'(1 + U'T)+.. "" u'(l+u'v)-
_ 31. — «. loco — — leg- -. - - -7^
y 1 — k'sin^' ¥ 1— .k'jiny'
- 95. — 10. 11. loco k leg. k'.
Ila. poilr. loco pro/teti leg. perfeetL
- 89. — 16. loco M teg. (— 1)^~M.
-47. — 10. Iow){...)Meg. (...J*.
8. deUnaom •yj -— - .
/^k" / . *Ki / . 8K\ / . 4(n-l)K\
j-^ .co..m(o)co..«.^ii.+ ^j».™^i. + — j.. co».m(u+-i— -i-j.
n W ■^ , jdiici
9. deleixlam i / — — , idiiciendum :
= /i,..a.<o,..,(. + 'J^)..„(..V")--»(-.+ ^^")-
Bb Digitized b, Google
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