I^unc igitur fractionibus sublatis prodibit haec aequatio: 



Xdx (x — y) — (33 x — yyd x — S-ydx (* y -+- *) — j^a«'(*+))i *— rt*' 

 — X3y(x- 7)— $dy — y xdy — Sxdy (jx+j) — t x xd y (x-+-yp *""* * 



Hujus ergo aequationis integrale completurn est ipsa illa aequa- 

 tio finita, quam supra sub triplici forma repraesentavimus, iu 

 qua littera oc est coristans arbitraria per integrationem ingressa^ 

 unde ex tertia forma integrale ita referri poterit 



a (2 A + y -+- 2 l(x + y) -+- »• (x -h /) 2 ) — A A (x— j) 2 — 2 A /3 (x-Hy) 



— aAyxy — 2A(fxj(x+y) — 2Aexx/;yH-/3/3 — 2 @ $x~y 



— 2 /3 e x j (x -+-/) -f- (W — y e) x xyy. 



sive 



XX (x— y) 2 — 2X(3 (*-+y) — •2X7»y-^-2X5'x>'('*-f-yj — z\zxxyy-{-$§ — z^Sxy 

 . — 2 (3e xjy(x-+>')-(-(5'5 — yi)xxyy' 



a — ' aX -+?-+■ 2 5 (x-f-.?0 -h £(*-*- j) 2 - 



$. 9. Quia in hac aequatione plures occurrunt litterae,. 

 scilicet A, /3, y, ^,- e, contemplemnr primo casus speciales, quibus 

 duae tantum litterae occurrunt, reliquis ad nihilum redactis. 



C a s u s I. 



quo 7 — <J =: e — O. 



§. 10. Aequatio ergo differentialis erit 

 A 2 x (x—y) — A dy (x — y) — /33 x — fidyzZO 

 sive A (x — y) (9 x — 3 j) — /3 (3 x -f- 3y) zb o 

 cujus integrale sponte se prodit 



A (x — j) 2 — 2 /3 (x -+-/) — Const. 

 Generalis vero integralis forma hoc casu praebet^ 



XX (x — y) 2 — 2 X ( 3 {x-\-y) -+-(3 (3. 



C* i i ' —————— - — — — — 



2X- 



