84. L. EULERI OPERA POSTHUMA. Mechanica, 



At ex prius integrata est Vp = '^"*""°^ — > unde illa aequatio abibit in hanc 



(siam .Vc-t-Vr)^ . , c — 6 (sin cp .Vc-*~Vr)i 



~ — - — — = xsinq)-i-o — c, erffo x = - h-^^ — f 5 — — 



CO8* 93 -^ »0 sin 93 8in 93 cos <p 



Hi valores pro yp et x inventi surrogentur in aequatione dr h- 2d(pVpr = xd^ cos g), atque orietur: 



, IdffiVcr 'irdf sin <p , (c — b)d(p cos 9 d?) (sin 95 ."/c -»- "/r)* 



cos 99 cos 9} sin q) sin 9) cos 9 



quae reducta praebet hanc; 



, rdm(sin^m — C08* ©) cdcp bdmcosm 



dr-i ^-^. — =- : • 



sm 9} cos 99 sin q> cos 99 sin tp 



33. Sic itaque pervenimus ad aequationem duas tantum variabiles r et g) continentem, quae 

 divisa per sin 9? . cos g) fit integrabilis ; erit enlm 



r e sin ffl c cos (p b cos cp . , 



= : — --i — : — ^-Hfl, ideoque 



8in 99 cos <p cos 99 sm 99 sin 99 * 



r = c sin^ (p-t- (b — c) cos^ (p-t-as\ncp cos g>. 

 Ponatur b — c = h^ atque per angulum (p omnes quantitates ita determinabuntar, ut sit 



Yr = cos (py{c tang^ cp-t-a tang (p-k-h)^ 



yp= — ^-i-sin9py(ctang^90-Hatang9?-«-/i) et 



2c tang 99 -H o -+- 2 "/(cc tang* 9» -*- ac tang 9 -+- cA) 



X^=z- — •' 



cos<p 



Ex his colligetur, (Fig. i\2.) curvam a corpore descriptam esse parabolam DAP, cujus vertex slt in 

 D, ex quo si ad horizontem OF perpendlculum BC demlttatur, erit OC = a, DC = hf et distantia 

 foci a vertice DE = c. Irratlonalitas evanescit, si fuerit c/i = -j aa, quo casu parabola per ipsum 

 punctum transibit. 



