OF ELLIPTIC FUNCTIONS. 
409 
whence also 
2 -2<P 
y — 0 2 + T 
Also 
4d 8 (0 2 _0+ 1 ) a =y 2 (0 2 + 1 ) 2 , 
wherefore 
2(P-Q+l) 2 =-d(Q 2 +l) or 2(3 2 -^+l) 2 + ^ 2 + l) = 0, 
or 
^ 2 +l) 2 +2(S 2 -fi + l) 2 =0, 
that is 
20 4 -30 3 +6S 2 -34+2=O, 
or finally 
(2tf-Q+l)(P-Q + 2) = 0. 
We have thus (20 2 +l) 2 =0 2 , that is 40 4 -f30 2 +l = O or 4& 2 +3#-}-l = 0, or else 
(9 2 +2) 2 =^ 3 , that is 0 4 -f-30 2 -f-4=O or # 2 -f3#-|-4=0 ; viz. the equation in Jc is 
(4£ 2 + 3£+l)(£ 2 + 3£+4)=0, 
these being in fact the values of Jc given by the modular equation on putting therein 0=1. 
The equation of the order 32 thus contains the factor -|(0, l) 4 } at least twice, and it 
does not contain either the factor 0—1, or the factor |(0, 1) 6 [ belonging to the quintic 
transformation; it maybe conjectured that the factor {(O, l) 4 } presents itself six times, 
and that the form is 
{(O, 1) 4 } 6 (0, 1) 8 =0 ; 
but I am not able to verify this, and I do not pursue the discussion further. 
22. The foregoing considerations show the grounds of the difficulty of the purely alge- 
braical solution of the problem ; the required results, for instance the modular equation, 
are obtained not in the simple form, but accompanied with special factors of high order. 
The transcendental theory affords the means of obtaining the results in the proper form 
without special factors ; and I proceed to develop the theory, reproducing the known 
results as to the modular and multiplier equations, and extending it to the determination 
of the transformation-coefficients a, j3 . . . 
The Modular Equation . — Article Nos. 23 to 28. 
7tK' 
23. Writing, as usual, q=e K , we have u, a given function of q, viz. 
/o ^+g^+g^ + g 6 - • 
,—^/2q l + ql+qa . 1 + ? s_ 
=^/2q*(l^q + 2q 2 -3q 3 +4:q*-6q 5 +9q ,i -12q 7 + . . .) 
=\Z2q*f{q) suppose; 
and this being so, the several values of v and of the other quantities in question are all 
given in terms of q. 
The case chiefly considered is that of n an odd prime ; and unless the contrary is 
stated it is assumed that this is so. We have then n -\- 1 transformations corresponding 
to the same number n -\- 1 of values of v ; these maybe distinguished as v 0 , v n v 2 , . . .v„ ; 
