434 
PROFESSOR CAYLEY ON THE TRANSFORMATION 
The unused equations then are 
w 14 (2ay -t-2a/3+/3 2 )= v\t 2 + 2s£ +2^), 
u 6 (y 2 + 2«a + 2ccl+ 2/3y + 2(31) = fl 2 (2ya + 2y£+ 2da + 2/3^ + 1 2 ), 
ir 2 (2ys+2cct;+2>yt+2l3e+2(3Z;+}> 2 )=v 2 (>y 2 +2o i e+2at;+2 7 ]>+2(3B+2(3l), 
u~ 10 (s 2 + 2y£+2Sa + 2B£)= v 2 (2«y + 2a&+2/3y + j3 2 ) ; 
hut I attend only to the first and last, which, it will be observed, contain y, o linearly. 
If in the first instance we substitute only for a, £ their values, the equations become 
^« 2 +( 3 )-$.(.+ 2 ^) 
®-'V 
say, for a moment, these are 
+ 
+w ,2 .2y —vu g . 2S=0, 
5-5 (*+«} • 2 5'+{5-5+“- ,!£ } • 2S =° • 
Here 
A.-] - ! 5 . 2 y-f-Q. 2^=0, 
B+R.2y+S .2&=0, 
1 : 2y : 2S=PS-QR : QB-SA : RA-PB. 
PS - QR =— + a - u" V + w 8 - «V(1 + j3) 
=i{ ^“+ v) — 2 w 1 V4-2m 8 -2mV— wV^j, 
where the terms containing ^ disappear of themselves, viz. this is 
= i(~-2u 10 v 2 +2u 8 -u 7 v 3 ^ 
= — 2v 3 u 3 — 2t;w — u 4 ) 
(observe that the term in ( ), equated to zero, gives the modular equation for the case 
n= 3). It thus appears that y and & are given as fractions, having in their denominator 
this function u i -\-2uv—2u 3 v 3 —v i . 
49. To complete the calculation, we have 
QB-AS=-to»(^ 2 -^/3 ! ) 
-{tf’/3(2+/3 )— % ,(,+2 £)}£-$+;?} ; 
viz. multiplying by 8, and substituting for 2/3, 2e their values, this is 
8(QB-AS)=-2«° [uV (m-$) 
“{““(a- 1 ) (jii+ 3 ) (m+"^)}(»i; _ 4+7' m) 
