66 REPORT—1846. 
Abel proposed to introduce three new functions, the first corresponding to 
that which he had previously designated by ¢9*. He now denotes it by Ab. — 
The second and third functions are apparently what the second and third 
kind of elliptic integrals respectively become, when, instead of x, we intro- 
duce the new variable §; 2 and @ being of course connected by the equation 
x=A6. The double periodicity of the function A and its other fundamental 
properties having been established, it was his intention to proceed to more 
profound researches. Some of his principal results are briefly stated. I may 
mention one, that all the roots of the modular equation may be expressed 
rationally in terms of two of them. 
One of the last paragraphs of the summary relates to functions very 
nearly identical with those which M. Jacobi discusses at the close of the 
‘ Fundamenta Nova,’ and which he has designated by the symbols Hand @. 
The second volume of Abel’s collected works consists of papers not pub- 
lished during his life. Two or three of these relate to elliptic functions. 
The longest contains a new and very general investigation for the reduction 
of the general transcendent, whose differential is of the form i P being, 
as usual, rational and R a polynomial of the fourth degree; together 
with transformations with respect to the parameter of integrals of the third 
kind. 
20. Having now given some account of the revolution which the disco- 
veries of Abel and Jacobi produced in the theory of elliptic functions, I shall 
mention some of the principal contributions which have been made towards 
the further development of the subject since the publication of the ‘ Funda- 
menta Nova.’ In Crelle’s Journal, iv. p. 371, we find a paper by M. Jacobi, 
entitled ‘De Functionibus Ellipticis Commentatio.’ It contains, in the first 
place, a development of the method of transforming elliptic integrals of the 
second and third kind, and introduces a new transcendent Q, which takes the 
place of ©, with which it is closely connected. M. Jacobi proves that the 
numerator and denominator of the value of y, mentioned above, and which 
have been denoted by U and V, satisfy a single differential equation of the 
third order. The remainder of the paper relates to the properties of Q (vide 
ante, note, p.55). When this function is multiplied by a certain exponen- 
tial factor it becomes a singly periodic function, and, which is very remark- 
able, its period is equal to one of the single or composite periods of the el- 
liptic function inverse to the integral of the first kind. By composite period 
I mean the sum of multiples of the fundamental periods. ‘The exponential 
factor being properly determined, its product by Q is equal to © multiplied 
by a constant. In considering this subject M. Jacobi is led to introduce the 
idea of conjugate periods. ‘These are periods by the combination of which 
all the composite periods may be produced. It is obvious that the funda- 
mental periods are conjugate periods; and there are, as may easily be 
shown, an infinity of others. 
In the sixth volume of the same journal we find a second part of the 
‘Commentatio.’ It contains a remarkable demonstration of the fundamental 
* In the ‘Précis’ Abel has adopted the canonical form of the integral of the first 
kind made use of by Legendre and M. Jacobi; so that the quantity under the radical is 
(1—2) (1—c? 2). It is worth remarking, that in his first paper in Schumacher’s Nachrichten 
this quantity is (1—e? x”) (1—c? 2”), while in the second it is the same as in the ‘ Précis.’ To 
this form he appears latterly to have adhered. 
+ It is not-clear whether by roots of the modular equation we are to understand the trans- 
formed moduli themselves, or their fourth roots, i.e. in M. Jacobi’s notation A or v. Vide 
supra, p. 50, 
a 
