4 M. Falk, On the Integration of partial 



, , 2iv /3 "if 3h ii''u 2iu\ , 7>v /Ti'-u 3?« 3-?i im\ 



where A = — - — -— — -=- + - H-t -,- — ., ^ -^] , 



«î/ vaaj'' ay axai/ ax) din \dy' ax axdy dyl 



whence B is obtained by interchanging u and v , 



1 ^ S?< ~bV ill 7)V 



dy dx dx dy 



Thus A , B and C do not involve (p , -^/ , cp' , -J/' . 

 From the expressions for p and q it follows, that 



^ ^F ^, 'èV 'èV ri ^F I , ^U 2)U 



C — (D^ ^ q — — p — , C %p' z= p -^ — (I -, • 



d(p dx dy dip dy dx 



Substituting from these the expressions for cp' and -^' in («), it is easy to 



see , that also (p and \|/ will disappear entirely , if — - — -. is independent 



TsF iF 



d<p dip 

 of them, or can by means of the given primitive be freed from them. This 

 has been the case in the foregoing examples. In the next following this 

 expression cannot be made independent of (p and ■v// , and the diiferential 

 equation Avill therefore be of the third order. 



Ex. 4. z = (p- 4^ — 4^' <p 1 



where for the sake of simplicity we suppose (p = (p(cr) , \// = %//(?/) . 



Here we have ^-f = 4^ (2cp - ^^) , ^-f^ = <f(v-2ip) , ^ =. 2(</- — 0), 

 d(f dip dtfdtp 



z'F 



whence 



dtpdw _ 2(y — i/') 



^F -iF (f'il'{ff—2(p)(2<f—ip) 

 d(f dip 

 and cannot be freed from <f and -^/ by means of the given primitive. 



Ex. 5. We return to ex. 3, but now we suppose its differential 

 equation of the third order. Substituting in (a) the expressions for ^' and 

 ^' from (i), we arrive at an equation of the form 



Är + /3s + 5.«= Ç + ;î j; (9^, V) (c), 



where a, , ß , y are the coefficients in [a) and , therefore , definite functions 

 of X 'Awd y] Ç and »? are in the same manner definite functions of a;, y , p 

 and q. As to F,^ (<p , ^>) , it neither is, nor can, by the aid of the given 

 primitive , be freed from <p and ip . 



