PARTICULARLY THOSE OE TWO VARIABLES. 
861 
69. Since 9{x t ) satisfies the general differential equation 
™-2x A 3 _M+2 KK '^-0 
^ 9 -2a,(K, ^ +2k„k„ dx -0 
dx, 
and the general term in fi>, so far as concerns x u , is a numerical multiple of 
d m 6(x u ) 
K„ 
dx u n 
it follows, exactly as in Section II., that 6? satisfies the r equations of the form 
dx 
?- 2 ^-|) S + 2 ^ S =°.( 245 )- 
That this is satisfied can be verified by means of the definition of <f >; and the same 
is true of the -Jr(r— 1) equations of the type 
d® , 2K S K< eP® A 
K t —+ —— t— r-= 0 
(246) 
dp s< t 7T 2 dx s dx t 
all satisfied by <P. 
70. Expressions for the constant terms in the even functions and for all coefficients 
in the expansions of all the functions in powers of the afs may be obtained as before. 
Noticing that 
(v, \ being either zero or unity, but not both unity at the same time) we have 
where 
r J\, \ 3 ,. . . , XA /2\| Ar 1 w ,*i 
u = - rvn-n k, s c,4c a 
W Vj,, v r j \7 t) V ° t=1 1 1 * 
r s=r t—r / ^2 
V 0 =coshl 2 $ t lo gpJj—r- 
1 *=it=i \dpApi 
, . . (247) 
. . . (248) 
and in the summation s, t are not to have the same value together. This gives the 
constant term in all those S r even functions in the characteristics of which no two 
corresponding numbers are unity at the same time. Similarly if we put 
