DOUBLE GAMMA FUXCTIOX. 
385 
Suppose that 
y z= (f) (x) satisfies the difierential equation 
y, y', ■ ■ ■ 2 /'"^) = 0, 
so transformed that it is rational and integral in y and its derivatives. 
Let the terms of class s be symbolically 
Ko(x) Q,o {y), Rj (a:) Q',( 2 /), ... , R^- (aj) Q/ {y) ; 
in terms of class (s — 1) being 
So {x) QVi( 2/),. , S/ (a?) QLi (y), 
and the functions R(x), S(x) being holomorphic. 
If ^(cc) satisfies the differential equation, <l){x) + i/;(a’|(yo) will satisfy the equation 
in which (x + Wi) is written for x and 4){x) + \fj{x\co{) the equation in which (x + wo) 
is written for x. 
Make the first substitution, divide the equations by Ro(^c) and Ro(fl7 -j- respec¬ 
tively, and subtract one from the other. We find 
Ato;Q - W-) + ^L.)] + . . + Q /+ -)] 
+ terms of lower class = 0 . 
But 
[<f){x) + xjj{x I ^ 2 )] — Q.° 
consists solely of terms of lower class than s. 
Hence either the equation which has been obtained vanishes identically, or we can 
reduce the equation for y to one in which there are fewer terms of class s. 
The equation cannot vanish identically uiiless the coefficients of the various terms 
of class s all vanish, which necessitates that the ratios 
Pt/i- (x) 
Po(*) Po(^) 
are doubly periodic functions of x of periods and ojo. 
The equation for y can thus be always reduced to one of the form 
R(a?)[po(x)Q,o(y)-f . . . 
+ So(cc)Q,"_i(y)-f . . . + S/(x)Q3_i(y) 
+ terms of lower class = 0. 
where all the coefficients are holomorphic functions, and, in addition, the functions 
p{x) are doubly periodic of periods and wg. 
VOL. cxcvi.—A, 3 D 
