ADDRESS. 31 
single variable, such as the elliptic sn, cn, or dn, but functions of p variables. 
Thus, in the case p= 2, which Jacobi specially considers, it is shown that 
Abel’s theorem has reference to two functions A(w, v), (uw, v) each of two 
variables, and gives in effect an addition-theorem for the expression of 
the functions \(w+w',v+'), \(u+w', v+v') algebraically in terms 
_ of the functions A(u, v), A, (uw, v), A(u’, v'), AY (wv, v’). 
It is important to remark that Abel’s theorem does not directly give, 
nor does Jacobi assert that it gives, the addition-theorem in a perfect 
form. Take the case p=1: the result from the theorem is that we have a 
function A(w), which is such that \(w +) can be expressed algebraically 
in terms of \(w) and A(v). This is of course perfectly correct, sn(w+ v) 
is expressible algebraically in terms of sn wu, sn v, but the expression 
involves the radicals /1—sn?v, /1—k’sn?u, /1—sn?v, /1—/sn2v; 
but it does not give the three functions sn, cn, dn, or in anywise amount 
to the statement that the sn, cn and dn w of w + v are expressible rationally 
in terms of the sn, cn and dn of w and of v. In the case p=1, the right 
number of functions, each of one variable, is 3, but the three functions 
sn, cn and dn are properly considered as the ratios of 4 functions; and so, 
in general, the right number of functions, each of p variables, is 4?—1, 
and these may be considered as the ratios of 4? functions. But notwith- 
standing this last remark, it may be considered that the notion of the 
Abelian functions of p variables is established, and the addition-theorem 
for these functions in effect given by the memoirs (Abel 1826, Jacobi 1832) 
last referred to. 
We have next for the case p=2, which is hyperelliptic, the two ex- 
tremely valuable memoirs, Gépel, ‘Theoria transcendentium Abelianarum 
primi ordinis adumbratio leva,’ Crelle, t. xxxv. (1847), and Rosenhain, 
Mémoire sur les fonctions de deux variables et 4 quatre périodes qui sont 
les inyerses des intégrales ultra-elliptiques de la premiére classe’ (1846), 
Paris, Mém. Savans Etrang. t. xi. (1851), each of them establishing on 
the analogy of the single theta-functions the corresponding functions of 
two variables, or double theta-functions, and in connection with them the 
theory of the Abelian functions of two variables. It may be remarked 
that in order of simplicity the theta-functions certainly precede the 
Abelian functions. 
Passing over some memoirs by Weierstrass which refer to the general 
hyperelliptic integrals, p any value whatever, we come to Riemann, who 
died 1866, at the age of forty : collected edition of his works, Leipzig, 1876. 
His great memoir on the Abelian and theta-functions is the memoir already 
incidentally referred to, ‘ Theorie der Abel’schen Functionen,’ Crelle, t. 54 
(1857); but intimately connected therewith we have his Inaugural Disser- 
tation (Gottingen, 1851), ‘Grundlagen fiir eine allgemeine Theorie der 
Functionen einer verinderlichen Complexen-Grisse’: his treatment of 
the problem of the Abelian functions, and establishment for the purpose 
of this theory of the multiple theta-functions, are alike founded on his 
general principles of the theory of the functions of a variable complex 
