548 
PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
which implies the relations 
b P -pfPl — 6(aP -(- € Q), 
dP-pJiQ=0(cP-p gQ), 
or, what is the same thing, 
(b-a6)P+(f-e0)Q=O, 
(d-c6)P+(h-g0)Q=O, 
and if the resulting value of P : Q be real, the last-mentioned equations give 
(ag — ce)& 2 — (ah -pig— cf— de)6 + lh — df= 0, 
and Q being known, the ratio P : Q is determined rationally in terms of l. 
320. The equation in 6 will have its roots real, equal, or imaginary, according as 
(aJi-P bg—cf— def— 4:(ag—ce)(bh—df), 
that is 
a 2 h 2 -piy-Pc 2 f 2 -Pd 2 e 2 
— 2ahbg—2ahcf— 2ahde—2bgcf— C lbgde—2cfde 
-piadfgh-pPbceh 
is = +, =0, or =— ; and I say that the transformation is subimaginary, neutral, and 
superimaginary in these three cases respectively. In the subimaginary case there are 
two functions Px-pQy which satisfy the prescribed conditions; in the neutral case a 
single function ; in the superimaginary case no such function. But in the last-men- 
tioned case there are two conjugate imaginary functions, Px-PQy , which contain as 
factors thereof respectively two conjugate imaginary functions UX+VY. 
321. Hence replacing the original x, y, X, Y by real linear functions thereof, the 
subimaginary transformation is reduced to the transformation x : y into &X : Y, where Jc is 
imaginary; and the superimaginary transformation is reduced to x-piy : x— iy into 
&(X+*Y) ; (X— ^Y), where Jc is imaginary. As regards the neutral transformation, it 
appears that this is equivalent to 
x=(a~P bi)X. + (c -P di) Y, 
y= ig+hi) Y, 
with the condition §=(ah-pbg) 2 — iagbh, =(ah—bgf, that is, we have ah—bg=. 0, or 
without any real loss of generality </=«, h=b, or the transformation is 
X— (a-p 5i)X-|- (c -p di)Y, 
y= (a+bi)Y, 
that is, #:^=X-|-£Y : Y, Jc being imaginary. 
322. The original equation after any real transformation thereof, is still an equation 
of the form 
(a,... Jx, yf= 0 ; 
and if we consider first the neutral transformation, the transformed equation is 
(a, ...XX+£Y, Y)"=0; 
this is not a real equation except in the case where Jc is real. 
