24] 
ON THE INVERSE ELLIPTIC FUNCTIONS. 
137 
{Fund. Nova, pp. 145, 133), by which © (u) is made to depend on the known function 
sin am. u. The other two numerators are easily expressed by means of the two 
functions H, ©. 
From the omission of Abel’s double factorial expressions, which are the only ones 
which display clearly the real nature of the functions in the numerators and denomi 
nators ; and besides, from the different form of Jacobi’s radical, which complicates the 
transformation from an impossible to a possible argument, it is difficult to trace the 
connection between Jacobi’s formulae; and in particular to account for the appearance 
of an exponential factor which runs through them. It would seem therefore natural to 
make the whole theory depend upon the definitions of the new transcendental functions 
to which Abel’s double factorial expressions lead one, even if these definitions were not 
of such a nature, that one only wonders they should never have been assumed a priori 
from the analogy of the circular functions sin, cos, and quite independently of the 
theory of elliptic integrals. This is accordingly what I have done in the present paper, 
in which therefore I assume no single property of elliptic functions, but demonstrate 
them all, from my fundamental equations. For the sake however of comparison, I retain 
entirely the notation of Abel. Several of the formula; that will be obtained are new. 
The infinite product 
(1). 
where m receives the integer values +1, ±2, ...+r, converges, as is well known, as 
r becomes indefinitely great to a determinate function sin — of x ; the theory of which 
might, if necessary, be investigated from this property assumed as a definition. We are 
thus naturally led to investigate the properties of the new transcendant 
x 
(2): 
M = ®IHI ( 1 + 
mco + nvi, 
m and n are integer numbers, positive or negative; and it is supposed that whatever 
positive value is attributed to either of these, the corresponding negative one is also 
given to it. i = V ( — 1), and v are real positive quantities. (At least this is the 
standard case, and the only one we shall explicitly consider. Many of the formulae 
obtained are true, with slight modifications, whatever &> and v represent, provided 
only co : vi be not a real quantity; for if it were so, mco 4- nvi for some values of 
m, n would vanish, or at least become indefinitely small, and u would cease to be 
a determinate function of ¿r.) 1 
Now the value of the above expression, or, as for the sake of shortness it may 
be written, of the function 
(3), 
1 I have examined the case of impossible values of co and v in a paper which I am preparing for Crelle's 
Journal. [The paper here referred to is [25], actually published in Liouville’s Journal]. 
C. 
18