DOUBLE GAMMA FUNCTION. 
387 
where g{x) and rix) are doubly fiinctions of x of periods and wo. And 
So(.r) 
S>i(x) 
(ic) ’ ‘ ’ E(a?) 
must be a function generated from the 
therefore one of the ratios 
function r 3 (xlc(jj, w^). 
The original equation therefore either contains the double gamma function 
implicitly among its coefficients, or it is reducible to the form (1). 
Continue our former procedure, and we see that either at least one of the ratios 
Tg(y) 
R{x) ’ 
E(a;) 
is composed of an additive number of equations of the type 
/(^ + "]) “/(■») = A (^) Qi 
f{x + Wo) - f{x) = 2 ^ (x) Qo {xp{x\oj^)] 
and is therefore generated from the double gamma function, or the original equation 
is reducible to one in which the ratios of terms of the three highest classes are doubly 
periodic functions of x of periods and co.,. 
The successive repetitions of the argument are now evident. Ultimately we 
reduce the equation to one in which either all the coefficients are doubly periodic 
functions (which is absurd), or to one in which the last term is generated from the 
double gamma function. 
Thus the ^proposition is established. The double gamma function cannot satisfy a 
differential equation in which the coefficients are finite combinations of, e.g., 
(1) Rational or irrational algebraic functions of a', 
(2) Simply or doubly periodic functions, 
(3) Simple gamma functions, 
(4) G functions, 
(5) Theta functions, 
or, in fact, of any functions which are not substantially reducible to or comijounded 
of the double gamma function itself. 
3 D 2 
