448 
PROFESSOR CAYLEY ON THE BICIRCTTLAR QUARTIC. 
16. I stop to consider the geometrical interpretation ; introducing dv, the formula 
may be written 
. R/(a? 2 +y 2 )& dv 
VH ’ 
dS= 
we have (x?-\-y 2 )R! =1 — ax— fiy— \/0, or 
(x^ + y*) R' l-ax-p y 
\ / 'x 2 + : 
is the perpendicular from the centre of the circle of inversion upon the 
tangent to the dirigent conic, and — - ■ is the half-chord which this perpendicular 
V '# 2 -)-^ 2 
forms with the generating circle. Hence - — —1 = (perpendicular — half-chord) 
half-chord, the numerator being in fact the distance of the element cZS (or point X, Y) 
from the centre of inversion : the formula thus is 
dS=± g ^dv, ' 
where h is the radius of the generating circle, g the distance of the element from the 
centre of the circle of inversion, and c the chord which this distance forms with the 
generating circle. If we consider the two points on the generating circle, and write 
<2S' for the element at the other point, then we have (^S + 6ZS ')=+^ — =21 dv 
2 C 
(which is Casey’s formula ds’ — ds= 2g d<p (273)). 
17. The foregoing forms of dX, d Y are those which give most directly the required 
value of tZS, but I had previously obtained them in a different form. Writing 
then 
this is 
A =pa.-cLy+{f—g)xy, 
xA=ft x 2 —ctxy+[(f+Q)x 2 -(g + G)y 2 ] ; 
(f+Qy=l-(g + Q)y 2 , 
xA=fix 2 -axy+[l-(g+ Q)(x 2 +/)], =y{ 1 - ax - (3y) + (x 2 +y 2 )({ 3 - (#+%), 
=(x 2 +y 2 ){yR'+p-(g-\-Q)y\+y*/U ; 
that is 
xA-ysJ £l=(x 2 +y 2 )\y~R' +p-(g+&)y\ 
and similarly 
-yA-x^Q=(x’+f){ x n'+a-(f+a)x\. 
