C a s u s II. 



quo /3 z: J z: ? ~ o. 



Hoc casu aequatio differentialis erit 

 hdx (x — y) — A <9 j ( x — y) — y (ydx -±- xd y) zzz o\ 

 cujus integrale pariter sponte se offert, quandoquidem erit 

 A (x — j) 2 — zyxy zzz const. 



Ex forma generali integrale fit a. zzr. — ^~~l\S^ ~- 



Quin etiam si fuerit tantum 8 zzz s zz o , qui fit 



C a s u s III. 



Aequatio differentialis erit 



A (x —- y) (dx — dy) — /2 (d X -f- d y) — y (y d X -f- X d y) ZZZ ; 

 cujus integrale est manifesto 



A (x — yf — 2 /3 (x -f- y) — 27 x y zzz const. 



Forma generalis autem praebet 



XX (jc — jy) 2 — a\$(xA-y) — 2X7 *,?-(- (3 (3 



a 2X-+-7 » 



ubi consensus est manifestus , sicque quoties ambae litterae 

 $ et e evanescunt , res nihil plane habet in recessu ; verum 

 si litterarum $ et e, vel altera tantum, vel ambae affuerint, 

 ejusmodi oriuntur aequationes differentiales , quarum integratio 

 per methodos usitatas non parum difficultatis involvit ; hujus- 

 modi igitur casus hic data opera evolvamus. 



C a s u s IV. 



quo /3 zzz y zzz e zzz o. 



§. 11. Hoc ergo casu aequatio differentialis erit 

 A (x — y) (dx — dy) — Syd.x (2 y -\- x) — ix dy (2 x -\-y) ZZZ o, 

 cujus integrale ex forma generali resultat 



XX (x — y) z — ^X Sxy^x-j-y) -\-SSxxyy 



a aX-i- 2 5 (x-+-y) * 



