516 L. EULERI OPERA POSTHUMA. CaU. integr. 



ut sit ' ,x\uv = V!^> iijJanlfiJa as eB>ifi*uI%dj/sin j/=prff cos rH-f/rfr sin r *»«i **» «^* »*«& = *^ -i- -t^Vj itjmtin .' 

 cujus integraliB ponainus p iih^i^-^^cciyr^-i— cbsy, aut (^os j/== g^cosr — |)sih r, essequ6 debeat*" ^ Nf-wj %::r: 



?,uuiiu9aijuij o?i*i3 t— -=. .('i •<(.:, 'n;!i;; ^ ' da m t>bmi , jir; \A> -f- ^ — — r:r ^'J 



" ¥^ " sm r . dp -i- cos r . «o = seu tano^ r = — • ' (««-<- 1)^ 



■^ ^ ° dp 



Deinde ob cos y inventum etiam sin y irinotescit , unde ex 'fec*ta iiy potl^est cos o sin y = p cos r ■+-q sin r obtinemiis 



p cos r -I- ff sin r ,„. . . ,, ., n • a . 



cos 6) =^ r : — —• (oi naec cum superionbus comparentwi videmus esse g = cos^, j)== — smacosy;, 



unde pulcnre sequitur 



-U aRtilnRlfp lea ?- aftliMflup tt> , Jbinu^jji^ p ^^ ^^^^ ^^^'° r - sia fl cos r cos y ,. j,,j|yj, „<j„ '^ ^j.^^ ^jlj ,^ 



"*' " ' sin y ■ 



j . - , , . . rlfibfjijoo'? rnhfiup q\>*\\ Jiii niubnBjadsiq ;jod f>A .Bolflid'. 



aemde etiam eleganter consentit valor > ' 



t) .tS-ic ' ai;! ■ , iitliifioq 36 ^fiot&idosle *s6JiJnB«p Jia non 



■■'■'•' - dq d0sinl9 ' ^ ' 



'.:!;:i.';i'r."> Tr^^^.^iiT. <^(s>i^cosv^) — - 



prorsus etiam ut supra.) 



Videamus nunc etiam quomodo iii*6 altera parte dx ratiociriium' prosequi debeat: Erat autem 



rfa; sin y = 6?r sin w, unde fit dx= — -. -? 



at jam invenimus 



siay s «9JcJi]iK;rtj> Jfi«?. or.ifp 



p cos r H- g sm r 



cosy^^-cosr — p&\nr et sin y = K (1 -i-^jpg^sm r cos r — jg^cos^^r — pp&mr) et cos o = : i 



iiib.p(\ GifB-idov' aiiotlanuJ '!Il o^q mitio sis ijq 'ibbM loiffitsad^ atuibc oiJu(o 



r { 1 — Tyj) ~* QQ^ ' dl* • 



unde sin « = -, — atque dx = -t- : ♦ (Consulamus iterum primam solutionem , ubi erat 



smy * y(l — pp — qq) ■ '■■ 



dx = d^~h-d<p eritque rf^)^—— -^^, jam autem mvemmus . =is« #iid-jd.;;: 



p . , y (1 —pp — qq) , , y (1 —pp — qq) 



consequenter MiidfiaailYrt i» noii 6^i ,9jijf»Jak4o, bq*. Jji ^»6 oilibnoa o'>sd loq 



•imjiJo: ''dtKMJit-y'^ '_, r P<^(? ^ _^ 



■ (1 — !??)i^(i — PFrr^g?),, 



quo valore substituto colligitur , 



j^ , ^ 1 / pdq dqddp \ 1 Ipdq^ -\-pdp^dq — dqddp -t- qqdq.ddpX 



'jx,,, «S — ^ — 9 — Y{i .- pp - qq) [l-qq'^ dq^-^dp^) ~~ V(l^pp-qq) \ . (1, ~ ?«) (^2* -♦- rfp^) j 



,Q ,- i a:;'i;> ; i'!"::'''^ ';'' :';'■!■ ■■ ■-■':')'•'. .i.':"T'*'i ii'>.\'--:: •. [ :,. . i^lHVi v.vkv \ -. 



quod novimus esse differentiale arcus | cuius cotanMns est .. . . , 



...'.'■/ ^udt^r^ oflwp ,i«J9i9bti*9b oic aiRj . 



•/(1 -p^^— ga) ' 



Quarum formularum evolutio nimis est diflicilis ) j^ -» .. 



"■■ ' ;:-.<Hfii'in i . .iiic ilr.i 



AnuD TENTAMEN. Quum essc debeat dx= —. — r- , ponamus dx,= d^ -\~ dcp , eritque 



J«\> — ■(•iif nitntaon r.fiLafdos'? 'moilBl' jfti! 



• " ,,,' - ar sm <y ■ , 



dt=zdx — afp = — : aqoj , 



• ^ '^ smy ^ 



ubi I et 9 sunt etiam arcus, quare dividatur per sin'^!, ut habeatur '*i>« rjjoijtri aithiiiiio 6U«<.tt ^^n^&mHl. 



:«»u.ii/-*i7i *ii; iii9UoiJJiWi5»'J6 iiit»ioa5*i*oq . ^| dri\no' ' df '^'^''^ ^ ***-^ "^^ *"^* ^** »\i flO" msJuB fiir 



quod ergo integrabile esse debet. In hunc finem statuatur .;i^, q ^^^3 



