OF ELLIPTIC FUNCTIONS. 
405 
12. I will first consider the cubic transformation ; here, writing for convenience 
|=3, the equations give 
lift 2 i 
20‘+i == OM 8 +20) 8 ’ that is, £¥'(0+2)— (20+l)=O, 
and 
k6 2 — O; 
the equation in 6 gives (^ 4 — l) 2 — 43 2 (^ 2 — 1) 2 =0, and we have thence 
£(0 2 — l) 2 — 40(M2— 1) 2 =0, 
that is 
£Q 4 - 4M 3 + 6H2 2 - 40 +k= 0, 
the modular equation ; and then # 2 3 4 — l+20(# 2 3 2 --l)=O, that is, O 2 — 1 +24(H1 — 1)=0, 
qz i 2a 
or 20=— which is = -j, say we have a = Q 2 — 1, 0=2(1— M2); consequently 
1 -y 1-x (O 2 — 1 + 2(£fl — l)a?) 2 
\+y~\+x {tl 2 -l-2(m-l)a?j’ 
. 1 a 2 + 2«/3 0 + 2 II 2 -4m + 3 
K— Qk, and M _ a2 — g — o 2 — 1 ’ 
which completes the theory. 
13. Keproducing for this case the general theory, it appears a priori that Q is deter- 
mined by a quartic equation; in fact from the original equations eliminating O, we 
have an equation 
U, V 
U', V' 
= 0 , 
where U, U', V, V' are quartic functions of os, 0 ; that is the ratio a : 0 has four values, 
and to each of these there corresponds a single value of O ; viz. O is determined by a 
quartic equation. 
14. Considering next the case n=5, the quintic transformation ; the elimination of 0 
gives the equations 
U V w 
TT'—V'—w’ 
where U, TJ 7 , &c. are all quadric functions of a, 0, y. We have thence 4 f 4 — 2*2, =12 
sets of values of a : 0 : y ; viz. considering a, 0, y as coordinates in piano , the curves 
TJV' — 11^=0, UW' — U'W=0 are quartic curves intersecting in 16 points ; but among 
these are included the four points U=0, U^O (in fact the point a=0, y = 0 four 
times), which are not points of the curve YW'-Y'W = 0 ; there remain therefore 16 — 4, 
= 12 intersections (agreeing with the general value (n-\- 1) . 2 iin ~ 3) ). Hence O is in the 
first instance determined by an equation of the order 12 ; but the proper order being =6, 
there must be a factor of the order 6 to be rejected. To explain this and determine the 
factor, observe that the equations in question are 
& 2 a 2 (2ay + 20y 0 2 ) — y 2 (2ay + 2a0 + 0 2 ) = 0, 
£V(a + 20) -y 3 (y + 20) =0; 
3 H 
MDCCCLXXIV. 
