~ 
62 REPORT—1846. 
¢,f and F in doubly and singly infinite continued products and series. They 
are derived from the expressions of @ @, &c. in terms of ¢ =, &c., by supposing 
m to increase sine limite, and are therefore analogous to the expression of 
sin ¢ in terms of ¢ which we have already mentioned. 
The second contains the development of what had already been pointed 
out with respect to the lemniscate, so far as relates to the division of its peri- 
meter by any prime number of the form 4m-+ 1. In an interesting note 
which M. Liouville communicated to the Institute in 1844, and which is 
published in the eighth volume of his Journal, p. 507, he has proved gene- 
rally that the division of the perimeter of this curve can always be effected 
whether the divisor be a composite or prime number, real or compleax (that 
is, of the form p + “—q, p and q being integers). In order to do this, it 
was only requisite to follow m.m., the reasoning by which Abel has shown 
that the equation which presents itself in the problem of the division of the 
circumference of the circle is always resoluble. Thus, as M. Liouville has 
remarked, his analysis is implicitly contained in Abel’s. 
This memoir also contains Abel’s theorem for the transformation of elliptic 
integrals of the first kind. It is equivalent to that of M. Jacobi; nor is the 
demonstration, though presented in quite a different form, altogether unlike 
M. Jacobi’s. 
Abel begins by considering the sum of a certain series of ¢ functions whose 
arguments are in arithmetical progression. He shows that the sum of this 
series is a rational function of its first term. If we call this sum (multiplied 
by a certain constant) y, and the first term 2, then y is such a function of ¢ 
as to satisfy the differential equation already mentioned, viz. 
va-yyd—vy) M V¥a=2)0—Ray 
or rather an equation of equivalent form. In fact y is m.m. the same func- 
tion of x that it is in M. Jacobi’s theorem. Thus the sum of the series of 
elliptic functions is itself, when multiplied by a constant, a new elliptic fune- 
tion, having a new modulus, and whose argument bears a constant ratio to 
that of the first term of the series. It appears also that for the sum of the 
elliptic functions we may, duly altering the constant factor, substitute their 
continued product. Thus, beside the algebraical expression of y, there are 
two transcendental expressions of it, both of which are given by M. Jacobi 
in the ‘Fundamenta Nova. At the close of the memoir Abel compares his 
result with the one in Schumacher’s Journal, No. 123, and mentions that he 
had not met with the latter until his own paper was terminated. 
19. In the 138th number of this journal Abel resumed the problem of 
transformation, and treated it in a more general and direct manner than had 
yet been done. This memoir appeared in June 1828. M. Jacobi, in a letter 
to Legendre, has spoken in the highest terms of Abel's demonstration of the 
formule of transformation: he says, “Elle est au-dessus de mes éloges, 
comme elle est au-dessus de mes travaux.” An addition to this memoir, 
establishing the real transformations by an independent method, appeared in 
Number 148 of the same journal. ‘These two papers are printed consecu- 
tively in the first volume of Abel's Works, pp. 253, 275. 
In the first of these two remarkable essays Abel makes use of the perio- 
dicity of the function ¢ 6, or, as he here denotes it, AG, to determine @ priori 
what rational function of 2, y must be in order that the differential equation 
dy dz 
Vie) (=F) VER8) Gea) 
