426 
PROFESSOR CAYLEY ON THE TRANSFORMATION 
1 — 7«(1 — uv) (1 — uv + u^v' 2 ) 
M u — v 7 
( 1 ) 
— 7 ^( 1 — UV) (1 —UV+VpV 2 ) 
v — u 7 
( 2 ) 
V 4 — l jv 4 (\ — uv)[\ — uv + ifiv 2 ) 
v 4 M u 3 (u—v 7 ) 
U 4 — 7 m 4 ( 1 — uv){\— uv+vPv 1 ) 
V 4 V 3 (v — u 7 ) 
(3) 
(4) 
so that here again the third and fourth forms are identical with the second and third 
forms respectively ; there are thus only two forms, and the elimination of M gives 
(u — v 7 )(v — u 7 ) + 7uv(l — uvf(l — uv + w V) 2 = 0 , 
which is a form of the modular equation. 
36. If in the foregoing equation, 
F'v.^=R(>, v ), 
we make the change u, v, ^ into v, +w, +wM, it becomes 
+ F'w. wM=K(v, +w); 
and combining these equations, we have 
±nW 
Fu R(w, + «) . 
’ F'v R(w, v) ’ 
or substituting herein the foregoing value 
M g 1 (1 — v 8 )vl?'v 
n (1 —u 8 )uWu 
this becomes 
_v(l-v 8 ) _R(v, ±u) +for w = 3 or 5 (mod. 8), 
4~m(1 ^^ 8 ) R(«, v) —for 1 or 7 (mod. 8), 
which must agree with the modular equation: thus in the last-mentioned case n— 3, 
where we have 
^F'v . ^ = 3(w V -\-2u 5 v -f 1 )u, 
or say 
E(«q v)= (wV+2w 5 'y-f-l)w, 
and therefore 
■ R(i>, — u)= (v 3 u 2 — 2uv 5 -{-l)v ; 
the equation is 
v(l— v s ) (v' 2 u‘ 2 —2uv 6 + l)v. 
4~m(1 — u 8 ) (i> 2 w 2 + 2u b v+ 1 )u 
which is right ; for Jacobi, p. 82, the modular equation, gives 
1 — u 8 = ( 1 — v?v*)(x?v? -f- 2 u 5 v + 1), 1— ?j 8 =(1 — wV)(?rV —2uv 5 -\-l). 
