272 L. EULERI OPERA POSTHUMA. A^ron.mtth- 



et 2dxddx H- 2drddr=kgdt- { '^''Z'"^'"' -^"^ ^(»*» "^ !><*/) (;!*- r»)) 



Ponamus nunc pro motu corporis M angulum JLM = g)^ distantia existente L>/=r, ut 

 £c = {• cos g) et j = (^ sin 9P , hincque ydx — xdy = — vidcp et dx^ -+- dy^ = dv^ -*- vvdcp'^. Porro 

 pro motu corporis iV statuatur distantia LiV=a, quae hactenus erat =J?, et angulus JLN=&j 

 ut sit r = M cos ^ et V} = usin&j hincque 



w = V({ucos& — f cos cp)^-i- {u sin »9^ — c sin 95)*) = V(uu — 2uv cos (^ — &)-*- cc) 

 et t}* — xp = uv sin (^ — d-)y atque r dcc -f- p d^ = «dc cos (<p — &) — uvdcp sin {cp — d-). 

 Unde nostrae aequationes erunt 



d. {i',dcp) = — 2gNui>de sin [cp — &) (^ — ^) 



d,{dv^-^vvdcp^) = kgde (Zl^^t^^—.^^ N{udv cos (9?—^) — upd^) sin (y—.^)) (^— ^,) 



seu si differentialia secundi gradus non rcformidemus, 



2dvd(p -+- vddcp = — 2gNudl'' sin {cp — ^) ^-i— -i) et 



^ 



ddc 



- .dy' = ^ 2 ., (£ -*- JW) ^ - 2i,A'd(' (^- « cos (y - *) (^ - J5)) , 



ubi u eX & tanquam quantitates per t datae sunt spectandae, terminique per N alFecti tanquam 

 valde parvi. 



Verum illae aequationes ad integrationem magis sunt pracparatae, et postcrior ob 

 wdw = vdv -H udu — udv cos (cp — &) — vdu cos (cp — &)-^ uv (dcp — dd) sin [cp — j^), 

 transit in hanc formam 



d . (dp^-H vvdq^^) =-^\g (L -+- M) tft»^ - kgNdl^ ^ d.co,(^-^)-.d^,in(,^^) >^ 



UfjJU^i / wdu — vdw C09 (y — ^) — uvd» sin (y — ^) — u>dtc \ 



nnde integrando quatenus licet obtinemus 



d.»-f- vvd^^= kg (L -*- i»/) dt' (D -*- 7) - ^^iVdi^ ^1L2^^ _ y vd^ .in^^-^) _^ 2 f^"^"'j^-^) ) 



-^kgNdt (-^-t-j ^^3 y ^^3 J, 



- <|(,»_H ('('(/y»^ kg (L-\-M) de- (D-*-l) -H 4^iVd/* ^^_ vcos(y-^) ^ 



sive 



u — i>cos(9J — ^) 2fcos(9) — ^) 



-♦-i^riVdf^yda^ 



^ *5r iN^dt* fuud& (1 ~ 1) sin [g> - i^ ). 



) 



