364 
PROFESSOR A. R, FORSYTH ON 
U+V+W= % 
p L 
=1 P'K) 
a direct proof of which has already been obtained. 
23. Combining (22) and (33) we have 
cdju)cd m { v) al m {u + v) 
. . ( 33 ) 
Therefore 
But by (7) 
IJ+V+W = — Nj = t =Jf—-al s (u) al s (v)al s (u + v). 
s= 1 P \ a s) 
m=p 
£ 
771= 1 
XV . tw 1 _ «=p 
pu m bv m ow m J m =i 
TN 1S tN ls 
—Sl^+-— 1 dV„ 
ou m bv m 
• (7') 
so that substituting for the Sir's and remembering that the Su’s and Sr’s are 
independent we have 
tU_tW_ SNj 
bu m 
bu m hw m 
bV bW 
bv m bw. m 
By the first of these 
t 2 U 
Sr 
b 2 W 
bil m bu n ' bw m bu\ 
S Wn 
h I± , 
bv m J 
6% 
bu m bu n n bu M bv 
$v n 
and therefore by (7') 
b 2 U 
+ 
Similarly 
b 2 Y 
+ 
t 2 W 
b 2 N 1 
bw m bw n 
falt m b'lC n 
SAV 
S 2 N a 
bw m bw n 
bu m bv n 
b 2 W 
bw m bw n 
bv m bv n 
b 2 W 
bw m bw„ bv m bu,i j 
from which we see that N x satisfies the series of differential equations 
(34). 
(35) 
_fc 2 N J ___S 2 N J _ 
bv, m bv n bv rn bu H 
of which there are bp{p — 1) in all. 
