XXIV
yEquaiiones art. XXII in casu nostro suppeditant -*
/7sin«L = -r-> cosu; = -7-9 — ncosù = m *-r-^ smv=m*-r-9*
^ dp ^ dq ^ dq ^ ^ dq
(dry /dry dr d^ dr do
fin = /î/i -- 4-3- 9 nn • 3- ~ -t- -r- • -r = o.
\ay/ \"/^/ "7 dq dp dp
Adjumento harum îequationum , quarum quinta et sexta jam in reliquis continenlur, séries evolvi potcrunt pro /*, ^, ^p, m, vol pro quibuslibet functionibus harum quantitatum , e quibus cas, quae imprimis attentione sunt dignae , hic sistemus.
Quum pro valoribus infînite parvis ipsarum />>, q fieri debeat rr=^ pp-\- qq^ séries pro r/'incipicl a lerminis pp -f- qq : terminos altiorum ordînum obtinemusper methodum coefficientium indcterminatorum {*), adjumento îccpiationis
\n dp I
-^ (-ir) =4/'/',
dp I \ dq
(*) Calculum, qui per nonnulla artificia paulhilum contrahi potest, hic adscribere superfluuni duxinuis.
scilicet
[,J rr = pp -^- l- f ''ppqq + ^,f'p»qq-^{{/"-^JT)p'çq -h elc.
Deîn habemus , ducenie formula /• siii df= 1— ?
^ 7.n dp
fa] r sin 4- = ;, - \ppqq - \f'ppqq - {{/" + hfV ) P'Çf -+" «te
- U°P7' — Wpp"/'
-{\h"-i-J"r)pq\ iiec non per formulam r cos ^ = ^ • — v- 9
[ 3J r,'os^ = q^\f'>ppq^kf'p'q + {\f" - -'-J-J " ) // q + etc.
Hinc simul innotescil angulus ^. Pcrinde ad computum anguli (f coucinnius evolvunlur séries pro rcosç atquc rsiny , quibus inserviunt acquationes differentiales partiales
li . r cos 9 . , . r/cp
— 7 — ^ = n cos cp . sin w — r sin ç • -r- » dp ' ' 'dp
d . rcos9 I . do ; — ^ = cos cp . cosu; — rsin o • -r-,
dff ' ' ^ dq
d . r sin 9 . . . dn
, = n sin Q . sm w — /• cos 9 • -r- 9
dp ' ' 'dp
d . r sin ». , ^o
-, = sm © . cos u^ — r cos ç • ■— <?
, dm , . do
n cos d/ • -r- -f- sin (L • -p- = o ,
^ rt/y ' dp
quarum combinatîo suppeditat
r sin >L d . r cos o , d . r cos 9
' • - - , — ^ H- rcosd/ — , =r coso,
n dp ' dq '
r sin ^L r/ . r sin V , r/ . r sin «
^L . . — J. -1-. /• oos u/ = r sin o.
n dp ' dq *
Hinc facile evolvuntur séries pro r cos (f , /'siii (f , quarum termini primi mani' festo esse debent p ex. q ^ puta
[4] rco,<f = p + \ppqq + -hf'ppqq-^{-hf"-i-J''f'>)p*qq-^eic.
(hh''—hf''r)pq\
[5] rs\u<f = q-krppq-\f'p'q-{-hf"--hPr)p'q-^tc.
— lë^ppqq—TiS'p^qq
-iih'^-hWrDppq'.
E combinalione aequationum [2], [3], [4]? [5J derivari posset séries pro /•/•cos (^ -h 9), atque hinc , dividcndo per scrîem [i], séries pro cos {^ -4- y), a qua ad serîem pro ipso augulo ^ H- 9 descendere liceret. Elegantius tamen eadein obtinetur scquenti modo. Differentiaudo sequationem primam et sccundam ex iis quce iuitio hujus articiili allatse sunt, obtincmus
. , fin , ^'^ .1 ^+
sin ^^ '-r ■+- n cos 9 • -7-^ -1- sm u; • --!- =r o ,
qua , combina ta cum hac
■A.î^^sintL.lî-
prodit
/i cos 7 • -7^ -+- sin ù . -;^ = o, * aq * dp
r sin i r//i r sm ^î; r/ ( -I^ -h 9 ) , di'^ -h 9)
• — -h ' --—, — ^ -+- rcos df • -^-^ — ^ = o.
n (if/ n dp ^ dq
Ex hac îcquatione adjumento mcthodi coefficientium indelcrminatorum facile elicicmus scricm pro :^ -f- 9 , si perpeiidimus ipsius terminum primum esse dcbere ^ tt, radio pro unitate acccpto, atcjue dénotante 21: perîpheriam cîrculi ,
[6] ^-^^^ = i^-fOpr,-lf'ppq-(if"-ifofo)pȍ^etc.
— g°pqq — 'js'ppqi
Opcr.T pretium videiur, cliam aream triangull ABD in serîem evolvere.
Huic evolutioni inservit îcquatio conditionalis sequens, quae e consîderatio-
nibus geometricis salis obviis facile derivaiur, et in qua S aream qusesitani
dénotât ,
rsin!/ f/S , dS rsinS >. ,
— ■ , — h /• cos i/ . — - = . //* da.
n dq ^ dq /i «^ ''
integrationc a ^ = o inoepla. Hiiic scilicet obtineiuus per mcthoduni coeffiiicnlium indeterniiuatorum ,
[7] ^ = \P^-T-.Pp'<1-nf'-p'<i-{T-J"- hfV)p'l-*-'i^-
- h^pq'- T^g'ppq-
-{■hh"-i-,rP)pq'.
A formulis arliculi pritîcccleiitis , qun» rcferuiitiir ad triaiiguluiu a liiieis brevîssîmis formalum rectaugulum , progredimur ad generalia. Slt C aliiid punctum in eadem liuca brevissima DH, pro quo, manenie />, charactercs q' , /•', y', ^\ S', cadem désignent, qua^ q^ r, (p, ^{^, S propunclo B. Itaoritur iriangulum înier puncta A , B, C , cujus angulos per A , B, C , latera opposita per a , 6, c, areamper t denotamus; mcnsuram curvaiune in punctis A , B, C resp. per a^ ^.y exprimemus. Supponendo itaque (quod lioet), quanlitates, p, q^ q — q^ esse positivas , habemus
Anle omnia aream t per sericin expriincmus. Mutando in [7J siiigulasquantitates ad B relatas in eas qurn ad C n^ferunlur, prodil formnla pro S', undo, usque ad quantitates sexti ordinis obtineinus
\ - Tis"" [q -+- q') C^pp -^4qq -^4qq' h- 4/v')
Hxc formula, adjumcnto scrici [aj, puta
csiii B = ;; (i — i/"^-/ — {/7»77 — ^g"7' — elc).
transit in sequcntcm :
(j = iacsinB! — ,\f' P (6 pp — 8 qq -h 7 qq' -h y (f' ff' )
-T-,g''{'ipPf-^^PPq'-fir-^-4qqq'-h^\qq'q'-i-iq")
Mensura curvatur.T pro quovis superficiel puiulo fit (per art. XIX ^ ubî m^ p^ q eraut qua* hic su ut u ^ q^ p)
elc.
I ddn if -+- 6 /^7 4- \7.hqq
n r/<y» I -\-fqq + etc.
^y '" — ^^ S^l — ( i a /* — 2^/) qq — etc.
Hiuc fil, quateuus />, q ad puuctum B referuntur,
/s= - ay »- ifp-6g^^q-if'pp - 6g>/7 - (i2/i«- 2/V«)9y -etc.,
fiec non
IntrcKlutmcIo tias ineiisuras curvaturic in soric pro a , obtincmus expressionem scquciitciu, usqiK> ad quaiilitates sexti ordinis (oxcl.) cxactam,
= ; «c sin H I -H TTï
