( 243 ) 
{ der Ary Ga A tere, cea 
when by 42) we understand Ge en) and likewise by A,?(A) 
A dr? 
Oe Dees AS 
Er ij el 
OL r=A=0; 
In this case it is only necessary that 4 is independent of g. The 
two intermediate integrals are found by determining the three 
independent integrals u,v and w of the system of simultaneous 
differential equations 
dez ede dp dy 
1 ed p En Ma.yverp) 0 
and by uniting these to 
uz fe) ¥ 
Vote AR 
Here 4 must be equal to pp (4) + w(r,g) and the two inter- 
mediate integrals are 
— | pdq N —f dq 
pe —{e wdg = fe), 
(1 — gp)y + 29 + J wdr = f(q)- 
We? 9 A 
Let U represent an arbitrary function of 7, y and z, then é must 
be equal to 
oU oU dU 
ET EE EN 0 
de Pf dy £ ap 
ee ad eho? UN 
— U aes = — }dzg— e-U LY) 4 
Dae tfen(S. sy tangy) © 7 We) 
where W denotes an arbitrary function of « and y. This result was 
already given by Goursat, “Equations aux dérivées partielles du 
second ordre’, II, p. 88. 
