534 
FUNCTION. 
[787 
we regard certain symmetrical functions of x, y, in fact, the ratios of (2 4 =) 16 such 
symmetrical functions, as functions of (u, v); say we thus have 15 hyperelliptic 
functions f(u, v), analogous to the 3 elliptic functions sn u, cn u, dn u, and being 
quadruply periodic. And these are the quotients of 16 double 6-functions 0 (u, v), the 
general form being 
0 (u, v) = ASS exp (a, h, b) (m, nf + mu + nv], 
where the summations extend to all positive and negative integer values of (ra, n); 
and we thus see the form of the d-function for any number of variables whatever. 
The epithet “hyperelliptic” is used in the case where the differentials are of the form 
just mentioned , where X is a rational and integral function of x; the epithet 
v X 
“ Abelian ” extends to the more general case where the differential involves the 
irrational function of x, determined by any rational and integral equation </> (x, y) — 0 
whatever. 
As regards the literature of the subject, it may be noticed that the various 
memoirs by Kiemann, 1851—1866, are republished in the collected edition of his works, 
Leipsic, 1876 ; and shortly after his death we have the Theorie der Abel’schen Functionen, 
by Clebsch and Gordan, Leipsic, 1866. Preceding this, we have by MM. Briot and 
Bouquet, the Théorie des Fonctions doublement périodiques et en particulier des Fonctions 
Elliptiques, Paris, 1859, the results of which are reproduced and developed in their larger 
work, Théorie des Fonctions Elliptiques, 2nd ed., Paris, 1875. 
14. It is proper to mention the gamma (I 1 ) or II function, r(w + l) = Tin, =1.2.3...n, 
when n is a positive integer. In the case just referred to, n a positive integer, this 
presents itself almost everywhere in analysis,—for instance, the binomial coefficients, 
and the coefficients of the exponential series are expressible by means of such functions 
of a number n. The definition for any real positive value of n is taken to be 
Г/г = I x n ~ l e~ x dx; 
J 0 
it is then shown that, n being real and positive, Г(?г + 1) = пГп, and by assuming that 
this equation holds good for positive or negative real values of n, the definition is 
extended to real negative values; the equation gives Г1 = 0Г0, that is, Г0 = oo, and 
similarly Г (— n) = oo, where -w is any negative integer. The definition by the definite 
integral has been or may be extended to imaginary values of n, but the theory is not 
an established one. A definition extending to all values of n is that of Gauss 
Tin = limit 
1.2.3 ... к 
?i + 1.№+2.w + 3 ... n + k 
k n , 
the ultimate value of k being = oo; but the function is chiefly considered for real values 
of the variable. 
A formula for the calculation, when x has a large real and positive value, is 
Tlx = ^2tt x x +? exp x + + ... j ,