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[24 
24. 
ON THE INVERSE ELLIPTIC FUNCTIONS. 
[From the Cambridge Mathematical Journal, t. iv. (1845), pp. 257—277.] 
The properties of the inverse elliptic functions have been the object of the 
researches of the two illustrious analysts, Abel and Jacobi. Among their most remarkable 
ones may be reckoned the formulae given by Abel (Œuvres, t. I. p. 212 [Ed. 2, p. 343]), in 
which the functions cf>a, fa, Fa, (corresponding to J acobi’s sin am . a, cos am . a, A am . a, 
though not precisely equivalent to these, Abel’s radical being [(1 — tfx-) (1 + e 2 # 2 )]*, and 
Jacobi’s, like that of Legendre’s [(1 — x 2 ) (1 — k^xf]^), are expressed in the form of fractions, 
having a common denominator ; and this, together with the three numerators, resolved 
into a doubly infinite series of factors ; i.e. the general factor contains two independent 
integers. These formulae may conveniently be referred to as “ Abel’s double factorial 
expressions” for the functions <£, f F. By dividing each of these products into an 
infinite number of partial products, and expressing these by means of circular or 
exponential functions, Abel has obtained (pp. 216—218) two other systems of formulae for 
the same quantities, which may be referred to as “Abel’s first and second single factorial 
systems.” The theory of the functions forming the above numerators and denominator, 
is mentioned by Abel in a letter to Legendre (Œuvres, t. II. p. 259 [Ed. 2, p. 272]), as 
a subject to which his attention had been directed, but none of his researches upon 
them have ever been published. Abel’s double factorial expressions have nowhere any 
thing analogous to them in Jacobi’s Fund. Nova; but the system of formulae analogous 
to the first single factorial system is given by Jacobi (p. 86), and the second system 
is implicitly contained in some of the subsequent formulae. The functions forming the 
numerator and denominator of sin am. u, Jacobi represents, omitting a constant factor, 
by H (u), © (u) ; and proceeds to investigate the properties of these new functions. This 
he principally effects by means of a very remarkable equation of the form 
/© ( w ) = 1/1 y? + BJ 0 du. f 0 du sin 2 am u,