FORMVLARVM DIFFERENTIALIVM. 199 



/N dx—fk vV^ - ' (a .V dy + ^y dx + x^^y^ y x dv\ ^y dx)) 

 -i-fx^j^ (3 dx + jt^^j" -'{nyxdj+{n + i) $ydx)) 

 — a. x^-^ ' y^ -\- y x^ '^'^ ~^ 'y^ "*" ". 



iS. PropoQta acquntionc diff.rcntiali liomoge- 

 nea R d x -h S d y ^ pro qua R ct S (unt fundiones 

 honnogeneac ipfarum x et 7 , inueftigcmus multi- 

 plicatorcm , qui eam intcgrabiicm re.ldat. Si igitur 

 ftatuatur hic multiplicator (p , habcbimus 



V z= Cp ( R + S />) , hinc 



M::r(J4)(R + Sp)-i-Cl)((J-|)-f-prf^)) 



N = (i|)(R+Sp)-HCp((5-^)4-p(^-f» 

 et P r:: S , tum vero erit 



Dum igitur ex hac aequatione valor litterae <t> erui 

 debet , primum obreruaffo iuuat, pro (p affumi poffe 

 functioncm homogcneam ipHirum x et y. Ponamus 

 autem Cp> cffe fundioncm homogeneam }ft dimenfio- 

 nis et per ea quae in Tom. I. Calculi Diflierent. 

 Jllurtris. Eulcrt §. 222. demonflrantur , habebimus 

 ;«(p = :c(^4)4-r(^f), hincque 

 Ky['^)^mK(^-Kx{f-) 



hoc igitur valore in aequatione fuperiori introdudo, 

 fiet 



f«R(p^(R^' + Sj)(^-|).::Cp(/(^5^)-i'(f|)) 



pro- 



