ON THE RECENT PROGRESS OF ANALYSIS. 55 
_ function of the modulus & which is usually denoted by g, and which occurs 
perpetually in this part of the theory of elliptic functions. If for the moment 
_ we denote this function by FA, so that g = Ff, then if for k we write ,, 
which we suppose to represent the modulus of the first real transformation 
of the mth order, we find that g" = F &,, so that if g, is the same function of 
_k, that qg is of k 
In = q" 
This singular property, and others of an analogous character, are of great 
use in establishing various formule *. 
Before discussing the evolution of integrals of the third kind, M. Jacobi 
has premised some important theorems. He proves that the elliptic integral 
of the third kind, though it involves three elements, viz. the amplitude, the 
modulus and the parameter, can yet be expressed in terms of other quantities 
severally involving but two. In order to this we introduce either a new trans- 
cendent t or a definite elliptic integral of the third kind, whose amplitude isa 
certain function of its modulus and parameter. It is almost impossible to 
tabulate the values of a function of three elements, on account of the enormous 
bulk of a table with triple entry; we therefore see the importance of the step 
thus made. M. Jacobi announced this discovery as generally true of elliptic 
integrals of the third kind, but his demonstration applies to that subdivision 
already mentioned, which was designated by Legendre “ Fonctions du troi- 
siéme ordre 4 parametre logarithmique,” and not to functions “4 parametre 
eireulaire ¢.” It is probable that this limitation was in M. Jacobi’s mind, but 
he does not seem to’ have expressed it. Further on, in the ‘ Fundamenta 
Nova,’ we find another mode of expressing integrals of the third kind in 
terms of functions of two elements, but this method also applies only to 
* fonctions du troisiéme ordre & parametre logarithmique,” the two methods 
being in fact closely allied. 
Legendre appreciated the importance of this discovery of M. Jacobi. He 
speaks of it in a letter to Abel, as a ‘ découverte majeure,” but adds that 
his attempts to extend M. Jacobi’s demonstration to the other class of intee 
grals of the third kind had been unsuceessful. The same remarks occur in 
his second supplement (Traité des Fonet. Ell., iii. p. 141). The distinction 
_ thus made between the two classes of integrals of the third kind appeared 
_ to Legendre sufficient to make it desirable to recognise in all four classes of 
elliptic integrals, so as to make the division between the two species of the 
_ third class coordinate with that between either and the first or second. 
_ Legendre says explicitly that M. Jacobi had announced, in making known 
__ his discovery, that it applied to functions “a parametre circulaire.” This 
i * A method of calculating elliptic integrals by means of g was suggested by Legendre, 
_ Yide Verhulst, p. 252, and M. Jacobi in Crelle. 
7 This transcendent is denoted by ‘, and is defined by the equation 
i dg 
ty f= 
y fi (00) Xe gy’ 
where E (¢¢)is the elliptic integral of the second kind, If we introduce the inverse nota- 
tion, and make 9 = am u,,we can readily establish the following result, 
1 . 
T= 50 = eff sin? am ud u?, 
_ The function Y, which is the logarithm of © (vide infra, p. 66), has many remarkable pro- 
 perties. ; i 
____ = In the former species (1 + 2) ¢ + =) is negative, and in the latter positive (vide 
_ ante,p.45). The specific names are derived from the circumstance that for the former the 
fundamental formula of addition involves a logarithm, for the latter a circular are. 
