PABTICULABLY THOSE OF TWO YAB1ABLES. 
825 
*,.a= 
Kx 
7 r 
9\1 
7T 
2 \* 
7T 
2^ 
7T 
2 \* 
7T 3 
" L x«d\ K* 
2! dp' 1 4! dp' 2 6! dp' 3 
■ /Tru a 2 d.fVK)* , d 2 .(«'K) 4 A 1 ® l,(«'K) ! 
(/cK) 3 —- 
(/cic'K 3 )^ 
2! dp' 4! dp' 2 6! dp' 3 
‘ d.( /cK) 4 ^ d 2 OK) 4 _a; 6 d 3 .(*K) 4 
' 2! dp' ' 4\ dp ,% 6! dpt' 2, 
a? 3 d.f/e/c'K 3 ) 4 . x* cP.fac'K*)* 
3! dp' 5! dp ' 2 
Expanding the cosines in the right-hand side of 
0o,o(~^ — H-2p cos -t+2 p* cos 2;r+2p 9 cos 3x-\- 
and equating coefficients of x 11 , we obtain 
d». K* 
<7/ 
= (27r) % {|)+4y+9y+16y 6 H- •..} 
which is easily deducible from 
/2KY 
\—j =i+2p+2y+V+ ■ ■ • 
and so verifies the above expansions. 
Similarly 
^^=(2ir) i {-p+ip- ( rp+Wp w — . . .} 
df >l 
dAjrcKy 
dp' n 
d n .(K K 'K s y 
9\ n 
=( 2 »)Mj;l>*+b) f'+ j p’I 
25 
dp' 
= ( 2 **) 
T.P*- 1 - 
9\« 
F+(f> 
1 
I 
> . ( 121 ). 
( 122 ). 
36. By means of the same formulae it is possible to obtain expressions for all the 
constant coefficients which arise in the expansions of all the S-’s in powers of x and y. 
Since by formula (4) 
\(x, y)=& 0 (-x,-y) 
it follows that 
^■o =c o 2"j(-^o.o> ®o,i> vY~^~ ■ ■ • 
4"^ 2n\ (^o»o> N 0) i, N 0l3 ,. . . , N 0>r , . 
• , n 0)2 ,x^ y)~ nJ r 
. . (123) 
d 2 ^ Q 
dx^dyf 
5 N 2 
where 
