1881.] solution of the equation x 3 + if - 1 = 0. 107 



are derivable from the formulae given, Legendre, Fonctions Ellip- 



tiques, 1. 1, pp. 185, 186, for the reduction to elliptic integrals of 



f dr 

 the integral R = - — — * , viz. Legendre writing 



J(l-z 3 ) 3 



and then ??i 3 = 2 and w 2 ?/ = 1 + a?, 



finds first 



.- f <?# 



and then writing r = v'H, a; = tan |#, and c 2 = | (2 - r 2 ), finds 



J Jl — c sin 9 

 we have therefore only to write sin <£ = sn u, to modulus 



and we thence obtain an expression for z in terms of the elliptic 

 functions sn u, en u, dn u. 



Writing x instead of z, and k for c, then 



And working out the substitutions, the resulting formulae are 



2r sn u dn u — (1 + en ^) 2 

 — 2r sn m dn w + (1 + en w) 2 ' 



in (1 + en u) {1 + r 2 + (1 + r 2 ) en m] 

 * " 2r sn w dn w + (1 + en w) 2 



where the modulus is & as above ; and these values give 



x* + f = l, 



dx — du . j 



{l-x° } r (l-yf> 



The verification is interesting enough; starting from the expression 

 for x, and for shortness representing it by 



A-B 



