( 620 ) 



Mathematics. — "On tlw dljj'erential cciuatiun of Monge" by Prof. 

 W. Kapteyx. 



(Commuiiicaled in the meeting of .lannary 30, 1004). 



If we suppose tliat in the differential equation 



Hr ^2Ks 4- Lt = () (1) 



H, K, L depend only on [> and q. the necessary and sufficient condi- 

 tions for the existence of two intermediate integrals are the following. 



In the first place H, K and L must he pro]K)rtional to the second 

 dilferential coeflicients of a function 6 ; thus 



^=A^A ,2) 



^11 ^12 ^32 



In the second place the function must satisfy the dilferential 

 equal ion of the fourth order 



2d{8, 

 = 3 



ö^i> dW d'l) 



V Op^ dp Oq Oq 



/dvy dndn /dTj\-' 



(3) 



where 



The general integral of this (Htfereiitial e(|iiation can be represented 

 b\ the fdUitwiiiU' formula»' : 



I' ~ 



q = 



II — V 



(^) 



= 2 fa' if' dn — 2 fh' x\'' dv — {a' — h') {if.' ^ if') 



C" — /') [if — »!'') - (<^ — ^') (."' — '^''i 



where u and r indicate arbitrary i)arameters and 



n — .1 {„) . h — h (r) , tp^if {") . If' = V' ('•) 

 four arbitrary functions, and 



(I z=z — , q ^^ — etc. 



I ' 7 " . ' 7 2 



da (In 



If the conditions (2) and (3) are satistied the eipiatiou '\ will 

 possess the two intermediate integrals 



z-zip"-xcf = 7,{n), 

 z — y y\)" — X A" zzz 3 (f), 



