432 
PROFESSOR CAYLEY ON THE TRANSFORMATION 
or observing that in this equation the coefficient of ^ is 
( 1 — mV) { 2 — 2 uv + 2 mV — 1 — mV } , 
=(1— mV)(1 —uv) 2 , =(1— uv) 3 {1 - \-uv ), 
the equation becomes 
(!-*>■) j^+|(l-w)*(l+M*>)+l-« 8 -4(l-«)(l+v)=0- 
45. This should be satisfied identically by the foregoing value of viz. it should 
be satisfied on writing therein 
1 7 u v — v7 
M 2 v u — v 7 ' 
1 7«(1— uv) (1 — WW + mV) . 
M u—v 7 
that is, we should have 
— 7 ^ (v — u 7 ){ 1 — v 8 ) — 1 4m(1 — uv)\ 1 + u 3 v 3 ) 
where observe that the — sign of the second term is the sign of the foregoing value of 
so that the identity being verified, it follows that the correct sign has been attributed 
to the value of 
46. Multiplying by v, the equation is 
— 7(1 — w 8 — 1 — uv)( 1 — v 8 ) — 14uv(l—uv) 4 (l -f mV) 
+ {1 — v 8 — 1 — uv}{ — 8(1— uv)-\-l — m 8 } +4(1 — uv)(v— u 7 )(u— v 7 )=0, 
viz. this is 
- 7(1 - u*)( 1-0 + 7(1 -uv)( 1-v 8 ) -Uuv{l-uv)\l + mV) 
+ (l-M 8 Xl-v 8 )-8(l-Mv)(l-'y 8 )+ 8(1 — m-u) 2 
— 1(1— uv)(l — u 8 )-\- 4(1 — uv){v— m 7 )(m— v 7 )=0. 
In the second column the coefficient of 1 — uv is 2 — u 8 —v s , viz. this is 
= (1 — m 8 )(1— v 8 ) + 1 — (uv) 8 , or it is =(1— uv) 8 -\-l — (uv) 8 . 
Reducing also the other two columns by means of the modular equation, the equation 
thus becomes 
— 6(1 — zty) 8 — (1 — uv){{1 — uv) 8 -\-l — {uv) 8 } — 14uv{l — uv) 4 (l + mV) 
+ 8 (1-uv) 2 
— 28mv(1 — mv) 3 (1 —uv + mV) 2 = 0. 
