1882.] Multiplication of Elliptic Functions. 273 



Denoting en a by x and en wa by y, where co = cos --- +isin — , 



o o 



an imaginary cube root of unity : then 



D, 



1 - y = P (1 •- a?) 



1 + .V = <? (» + «) -h D, 



y + ic = R (x — ic) -~ D, 



y-ic = S(l+x) +D; 



so that y can be expressed in terms of x by means of a linear 

 transformation, viz. 



_ co (k' + kx) - k' (1 - x) 

 y ~w (kT+kx) + k(l-x) ' 



where c = 2 + <s/3 ; and P, #, P, S are determined from the con- 

 ditions that x = 1 and y = 1 simultaneously when a= ; or ic = 0, 

 y = \Atc simultaneously when a — K. 



This linear transformation may be identified with numbers III., 

 IV., V. and VI. of Abel's linear transformations (Abel, QZuvres 

 Completes, t. I., p. 5G9). 



Conversely, these algebraical relations between x and y lead 

 to the differential relation 



dy codx 



Vl(i - f) if + O] Vl(i - «?) (*' + C)\ 



In the reduction of the integral 



<f> (z) dz 



/ 



(z A -l)<J(z*-b 3 )' 



considered by Legendre (Fonctions Elliptiques, t. I. Chap, lxvi.) 

 the modulus of the elliptic functions is sin 15°, and elliptic inte- 

 grals of the third kind are introduced, in which it may be proved 

 that the parameters, according to Jacobi's notation, are of the 

 form a, aa, (o' 2 a; where &y' = 1. [Quarterly Journal of Mathematics, 

 Vol. xviii. p. G6.) 



ft 7- ' 



Next, suppose -j? = \/7 ; then, denoting the modular angle, 



sin 26 = a, and c = -r-=cot0 = 8 + 3V?, 



