PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF 



If 



u (a, b, c,f, g, It, I, m, n, v, x, y, z) 



be an integrable combination of these equations, then we must have 



tin da 

 3 dx 



da 





du dh 



\ i olt i 



-f ?ip) + -^ (a 



3 r/A 3? 



a// </// a// 



.', H - e 



on 



(g 

 a 



= 0, 



,, 



( h 



Multiply the latter by '- , and add to the former; where p and q occur in 



the result, it is in the form 



.' H - } G - 00 



and so the equation is 



da 



36 



CK, 



Cli -, 



8c 

 AH - 



A 



(<> 5 , 3// a 



57 o/ + -5- So + --T- 

 fV 09 oA 



+ 



A 



i G - 

 A 



3, n 



j;II- 

 A 



1H ^ 3 

 - 

 A oy 



iQ- 

 A 



This must be satisfied in virtue of the three preceding equations and independently 

 of the actual values of 8, . . ., Bf. Substitute for S, 86, Sc ; then the coefficients 

 of Sy, ?>g, 8/i, and the term independent of these must vanish. We thus obtain four 

 linear homogeneous partial differential equations, viz., 



A 8a A H + 36 



3(6 I G + 00 3. A 3 



3 A 3a -.V G + 00 oc 



: a/ " ' "i H + 36 " " J~G + 00 a c 



i H + I G + 00 



, 7 a 3w , 



+ 4 x + x, + 



dn 



+ 



7 p 



A \ t)/ 



iG 00/ 



"- 



w , 



+ wi XT + ^ ^7 + ^ ^ + 



' 



