OF ELLIPTIC FUNCTIONS. 
407 
The thing to be observed is that the extraneous factor (Q — l) 6 , equated to zero, gives 
for O the value 0=1 corresponding to the transformation y=x of the order 1. 
17. Considering next n= 7, the septic transformation ; we have here between a, 0, y, £ 
a fourfold relation of the form 
( U, V, W, Z )=0, 
I XT', V, W', Z' | 
where, as before, U, U', &c. are quadric functions, and the number of solutions is here 
8 . 2 2 , = 32 ; to each of these corresponds a single value of 0, or $ is in the first 
instance determined by an equation of the order 32. But the order of the modular 
equation is =8; or representing this by {(0, 1) 8 }=0, the equation must be 
(Q, 1) 24 {(0, 1) 8 }=0, 
viz. there must be a special factor of the order 24. 
18. Oneway of satisfying the equations is to write therein cc= 0, c5=0 ; the equations 
thus become 
£0 2 =Oy 2 , 
y 2 +20y=m:(2,3y+0 2 ); 
or putting 0, y=a!, 0', 
yhs' 2 =O0' 2 , 
0' 2 + 2a'0'=O£(2«'0'+«' 2 ), 
which (with a', 0' instead of a, 0) are the very equations which belong to the cubic 
transformation; hence a factor is |(0, l) 4 }. 
Observe that for the values in question «=0, £ = 0, P=0'# 2 , Q—ct!, 
(P+Cb) 2 =^ 2 (a'±0^) 2 , =tf 2 (P'±Q'tf) 2 , if P'=a', Q'=/3', 
and therefore 
1—y _ \-x / T' -Q' x\ 2 
1 + y 1 + a? i^P' + Q!x ) 5 
which is the formula for a cubic transformation. 
19. The equations may also be satisfied by writing therein y=#a, !$=#0; in fact 
substituting these values, they become 
£V=Q£ 2 0 2 , 
2£V+A(2a0+ 0 2 )= 2a0)+ 2QA0 2 , 
^ V + 2#(0 2 + 2a0) = 2 Q# 2 (a 2 + 2a0) -h Q#0 2 , 
£ 2 (0 2 + 2a0) = Q,k 3 (a 2 + 2 «0) ; 
the first and last of these are 
£a 2 = O0 2 , 
0 2 + 2a0=O&(« 2 +2a0), 
which being satisfied the second and third equations are satisfied identically ; and these 
are the formulae for a cubic transformation; that is, we again have the factor {(O, l) 4 }. 
3 h 2 
