178 On the Transformation of Elliptic Functions. [Mar. 7, 



equations determine (3, 7 and the modular equation. If instead of (3 we 

 introduce M, by means of the relation -^=1 + 2(3, that is 2(3=^-— 1, 



then the last equation gives 2^{=u 3 v4~—^\ ; and a, /3, 7, 8 having 



these values, we have the residual two equations 



u\ 2a 7 + 2a(3 + (3 2 ) = v\*f + 2 7 S + 2/35), 



7 2 + 2/37 + 2a£ + 2/3$=: V(2a 7 + 2/3 7 + 2a5 + /3 3 ) , 

 viz., each of these is a quadric equation in ; hence eliminating i 



we have the modular equation ; and also (linearly) the value of ^ and 



thence the values of a, (3, 7, 8 in terms of u, v. 



Before going further it is proper to remark that, writing as above 

 o=al, then if 6=67, we have 



1 -j&e + 7.13 2 - oa 3 = (1 -fix) (1 + 7a 2 ), 

 1 + $x+ H x 2 + ^ 3 = (1 + i3x) (1 + 7^) , 

 and the equation of the transformation becomes 



i+?/~i+Ai+pv 



2/ 



viz., this belongs to the cubic transformation. The value of (3 in the 



cubic transformation was taken to be p= — , but for the present pur- 



v 



pose it is necessary to pay attention to an omitted double sign, and 



write /3=±— ; this being so, S=/fy, 'and giving to 7 the value T'W 4 , 5 







will have its foregoing value =— . And from the theory of the cubic 



equation, according as ft=— or = — — , the modular equation must 



v v 



be % 4 -^+2^(l-%V) = 0, or u i -v i -2uv(l—u 2 v 2 )=0. 



We thus see d priori, and it is easy to verify, that the equations of 

 the septic transformation are satisfied by the values 



o==l, (3= —,7= u\ c = — , and w 4 — « 4 + 2z^(l— u 2 u 2 ) = ; 



V V 

 1 7 



a=l, (3=— — , 7=— zt 4 , 6= — , and -zt 4 — v i — 2uv(l — it 2 v 2 ) = ; 



v v • 



and it hence follows that in obtaining the modular equation for the septic 

 transformation, we shall meet with the factors u*—v i ±2iiv(l—u 2 v 2 ). 

 Writing for shortness uv = Q, these factors are n 4 — u 4 d= 20(1 — 2 ), the 

 factor for the proper modular equation is u 8 + v 8 — ®, where 



6 = 80 - 28<9 2 + 5 6(9 3 - 70<9 4 + 5 60 5 - 280" + 86> 7 



