408 
PEOFESSOE CAYLEY ON THE TEANSFOEMATION 
Observe that for the values in question y=#a, we have P=a(l4-&r 2 ), 
Q=0(l+&r ! ); so that, writing P'=a, Q'=0, we have for y the value 
l-y __ (l— g)(F-Q'g) 8 
l+y — (1 + a?) (P'-t-Q^) 2 ’ 
which is the formula for a cubic transformation. 
20. It is important to notice that we cannot by writing a=0 or S = 0 reduce the 
transformation to a quintic one; in fact the equation k 3 a?=i 2& 2 shows that if either of 
these equations is satisfied the other is also satisfied ; and we have then the foregoing 
case a=0, cS = 0, giving not a quintic but a cubic transformation. 
And for the same reason we cannot by writing a=0, 0 = 0, y=0 or 0 = 0, y=0, c>=0 
reduce the transformation to the order 1. There is thus no factor 0—1. 
21. As regards the non-existence of the factor O — l, I further verify this by writing 
in the equations 0=1 ; they thus become 
&V=& 2 , 
&(2ay + 2a/3 + 0 2 ) = y 2 + 2y& + 20S, 
y 2 + 2 0y + 2a&+20&= jfc(2ay + 20y+ 2aS+0 2 ), 
& 2 + 2yS=£ 3 ( a 2 + 2a0), 
which it is to be shown cannot be satisfied in general, but only for certain values of k. 
Reducing the last equation, this is yh=k 3 u(3, which, combined with the first, gives 
ay=0§ ; and if for convenience we assume ct= 1, and write also 5= ir\/k (that is k=Q 2 ), 
then the values of a, 0, y, § are a=l, 0 = y3~ 3 , y=y, ^=0 3 ; which values, substituted 
in the second and third equations, give two equations in y, 6 ; and from these, eliminating 
y, we obtain an equation for the determination of Q, that is of k. In fact the second 
equation gives 
0 2 (2 y + 2y&- 3 +y 2 S- 6 )=y 2 +2y^ 3 + 2y ; 
or, dividing by y and reducing, 
y(l-^)=2f(fl 2 -l)(^-Hl), that is 
y(l + 0 2 )=-20 3 (0 2 -S + l), 
or, as this may also be written, 
( 7 +fi 3 )(l+6 2 )=-^-I) 2 , 
that is 
(r+^ 3 )= 
-0 3 (0-i) 2 
0 2 +l 
Moreover the third equation gives 
that is, 
y 2 + 2y 2 r 3 +2^ 3 4-2y=^(2y+2y 2 r 3 +2^ 3 + y 2 r 6 ), 
y 2 (0 4 — 2#-|-20 — 1) — 2(y+0 3 )0 4 (0 2 — 1)=0 ; 
or dividing by (P— 1, it is 
y 2 (0-l) 2 =20 4 (y-M 3 ); 
