PARTICULARLY THOSE OF TWO VARIABLES. 
821 
2Kx 
Writing u for —— and taking logarithmic differentials 
2K d0 o Q (u) 
p sin 2x 
p 3 cos 2x 
4 0 o ,o( u ) 
7r 
du 
1 + 2 p cos 2x +p 2 1 + 2'p* cos 2x +p 6 
p p s 
= sin 2x 
p 5 
1 + 2 \p cos 2x +p 2 ' 1 + 2p s cos 2x +p 6 ~ 1 + 2 p a cos 2x +p 10 
Differentiate again with regard to x, and then put x zero : 
\*/2IC 
\ 3 d A),o( U ) O 
p 
f 
1 & 1 
/ V * / 
i du o 3 
lo+py 
h (l+p 3 ) 2 
h (l+p 5 ) 3 ' 
But by (100) 
d^U x) _ / a d0Q, o(Q) 
dx 0 2 ** d* 
1 rfK 
27tK die 
Hence 
*vc' 2 K (ZK _ p p s . p’° . p 7 . 
2tt 2 ^' _ '(l+p) 2+ (l+p 3 ) 2H "(l+p 5 ) 3 " t "(l+p 7 ) 2 " ,_ 
so that now all the coefficients in (102) are known explicitly in terms of p. 
On the constants. 
S3. From the definitions of & 0 , c 0 we have 
c 0 =l + 2p+2pM-%> fl + . . • +2g+2^+V+ * • • 
+2p {?(r 2 +'j+2*(r 4 + ^)+q s (p +^)+ ■ • • 
+2 p 4 
1 K+S +2 4 K +3 +2V la +^ + • •• 
+ 2p 9 {?(»- 6 +^)+2 1 (» ,li +^)+2 9 (’ , ' 8 +^i)+ • ■ •} 
+ . . . 
with similar series for c lt c 2 , c 3 , c 4 , c 6 , c 8 , c 9 , c 13 , c 16 . Substituting these in the complete 
set of sixty equations of which (41) are the type, there will result algebraical identities 
in three quantities p, q, r (mutually independent), corresponding to identities in the 
“ cpseries ” in elliptic functions. As an example 
