58 REPORT—1846. 
Ido not presume to compare the merits of these two mathematicians. 
The writings of both are admirable, and may serve to show that if ever the 
modern method of analysis seems to be an éurepia rather than a réyvn, it 
does so, either because it has not been rightly used, or because it is not duly 
understood. 
To obtain a general view of Abel's writings it may be remarked, that his 
earliest researches related to the theory of equations. Of the ideas with 
which he was then conversant he has made two principal applications. The 
one is to the comparison of transcendents in the manner already described ; 
the other to the solution of the equations presented by the problem of the 
division of elliptic integrals. The second of these applications is contained 
in the memoir published in the second volume of Crelie’s Journal. 
He begins by introducing an inverse notation ¢ (w) corresponding to the 
function denoted in the ‘ Fundamenta Nova’ by sinamw, while f(w«) and 
F (w) correspond respectively to cosam «% and A am w. This notation 
has the defect of appropriating three symbols which we cannot well spare. 
On the other hand it is certainly more concise than M. Jacobi’s. 
He then verifies the fundamental formule for the addition of the new 
functions, and goes on to show that they are doubly periodie*. He next 
considers the expressions of ¢” a, &c. in ga, &c., and proceeds to prove 
the important proposition that the equation of the problem of the division 
of elliptic integrals of the first kind is always algebraically soluble. 
In order to illustrate this, which is one of the most remarkable theorems 
in the whole subject, it may be observed, that as any circular function of a 
multiple are ean be algebraically expressed in terms of circular functions 
of the simple arc, so may ¢na, fna, Fna@ be algebraically expressed by 
means of 6a, fa, Fa. 
Conyersely, as the determination (to take a particular function) of sin a in 
terms of sin x @& requires the solution of an algebraical equation, so does that 
of ¢a in terms of gma. The equation which presents itself in the former 
case is, as we know, of the mth or of the (2)th degree as n is odd or even. 
But the equation for determining ¢ @ rises to the (®)th degree in the former 
case, and in the latter to the (2 x*)th. We may however confine ourselves 
to the case in which 2 is a prime number; since if it be composite the ar- 
gument of the circular or elliptic function may first be divided by one of 
the factors of n, and the result thus got by another, andsoon. Thus setting 
aside the particular ease of m = 2, we shall have to consider, in order to 
determine sin @ or  @, an algebraical equation of the mth or (n®)th degree 
respectively. 
In consequence of the periodicity of sin a, the roots of the equation in 
sin @ admit of being expressed in a transcendental form; they are all in- 
cluded in the formula sin { @ + 2p"), in which p is integral, and which 
n 
therefore admits of only different values. 
* The formule in question differ from those already given, only because Abel’s form of 
dg 4 
iptic i i , which becomes the same as Legendre’s on 
the elliptic integral isf- Go ea) peat) 8 
making e? = —1, The double periodicity of the functions is expressed by the formula 
96=of{(-1)"t" 64 motnoV—)}) 
with similar formule for f and F. The quantities m and n are integral, and 
1 d zt 
al dx ras 
- ig "V0 = eat) (eat) tag 4 ip VW (1 + c3a%) (1 — a)’ 
