38 G 
MR. E. M'. BARXES OX THE THEORY OF THE 
Divide the equation by Pv(x) and subtract it from the equation which results from 
changing x into x (d^ We obtain 
+ 
Sn(.2; + wd 
ii{x + +^(^ 1 ^ 2 )}- 
+. 
+ 7Y (a;)[Q/{ ^(^’) + ^{x\ CO.) } 
- Q/[</>(^)}] 
^o(D /^o 
+ h 
R {x + 0 ) 2 ) 
R(v) 
+ terms of lower class = 0. 
This equation will not vanish identically unless the functions 
Sd.D S/Dd 
R(D 
. , all satisfy relations of the form 
Rf-r) 
f{x + ^ 2 ) —f{x) = S p^a;) Q 2 ^[i//(a;-l W. 2 )] 
i- = 0 
and therefore l)y symmetry relations of the form 
f{x + Wo) -f{x) = X iH- (x) Qi^' 
A -=0 
ip(xlo)i) 
The quantity (?? + 1) which gives the number of terms on the right-hand side of 
these two relations will not vanish unless the original equation can be reduced to the 
form 
(*0 [2h,0 (^) Qs (y) + • • + (•^) Q' (,v) 
+ Ps.o (^) Q^i 0/) + • ■ . /(.x) Q^-1 (y)] 
+ To (x) Ql, (y) + . . . + T„, (x) Qn 0/) 
-\- terms of lower class = 0.(1), 
where the coefficients are holomorphic functions of x and the p’s are doubly periodic 
functions of periods Wj and oj.. 
Either then the original equation can Ije reduced to this form, or at least one of the 
ratios 
R(,r) ’ 
is composed of an additive numl)er of solutions of difference 
equations of the form 
r f{x + wj) -f{x) = p (x) xfj (a; I Wo) 
\f{x -f Wo) -f{x) = p {x) xfj (a'jwi) 
But the most general solutl(m of such a })air of equations is 
<2 {x) <ji (a:) -f r [x) 
