30 REPORT—1883. 
dn of « +y are fractional functions having a common denominator: this 
is a reason for regarding these functions as the ratios of four functions 
A, B, C, D, the absolute magnitudes of which are and remain indeter- 
minate (the functions sn, cn, dn are in fact quotients [@,, ®,, @3] + O, of 
the four theta-functions, but this is a further result in nowise deducible 
from the addition-equations, and which is intended to be for the moment 
disregarded ; the remark has reference to what is said hereafter as to the 
Abelian functions). But there is in regard to the functions sn, cn, dn 
(what has no analogue for the circular functions), the whole theory of 
transformation of any order » prime or composite, and, as parts thereof, 
the whole theory of the modular and multiplier equations; and this 
theory of transformation spreads itself out in various directions, in 
geometry, in the Theory of Equations, and in the Theory of Numbers. 
Leaving the theta-functions out of consideration, the theory of the proper 
elliptic functions sn, cn, dn is at once seen to be a very wide one. 
I assign to the Abelian functions the date 1826-1832. Abel gave 
what is called his theorem in various forms, but in its most general 
form in the ‘Mémoire sur une propriété générale d’une classe trés- 
étendue de Fonctions Transcendentes’ (1826), presented to the French 
Academy of Sciences, and crowned by them after the author’s death, 
in the following year. This is in form a theorem of the integral 
calculus, relating to integrals depending on an irrational function y 
determined as a function of 2 by any algebraical equation F(a, y) =0 
whatever: the theorem being that a sum of any number of such integrals 
is expressible by means of the sum of a determinate number p of like 
integrals, this number p depending on the form of the equation F(a, y) =0 
which determines the irrational y (to fix the ideas, remark that con- 
sidering this equation as representing a curve, then p is really the deficiency 
of the curve; but as already mentioned, the notion of deficiency dates only 
from 1857): thus in applying the theorem to the case where y is the 
square root of a function of the fourth order, we have in effect Legendre’s 
theorem for elliptic integrals F¢+ FW expressed by means of a single 
integral Fy, and not a theorem applying in form to the elliptic functions 
sn, cn, dn. To be intelligible I must recall that the integrals belonging 
to the case where y is the square root of a rational and integral function 
of an order exceeding four are (in distinction from the general case) 
termed hyperelliptic integrals: viz.,if the order be 5 or 6, then these are 
of the class p =2; if the order be 7 or 8, then they are of the class p =3, 
and so on; the general Abelian integral of the class p=2 is a hyper- 
elliptic integral: but if p=3, or any greater value, then the hyper- 
elliptic integrals are only a particular case of the Abelian integrals of 
the same class. The further step was made by Jacobi in the short but 
very important memoir ‘ Considerationes generales de transcendentibus 
Abelianis,’ Crelle, t. ix. (1832): viz. he there shows for the hyperelliptic 
integrals of any class (but the conclusion may be stated generally) that the 
direct functions to which Abel’s theorem has reference are not functions of a 
