446 
PROFESSOR CAYLEY ON THE BICIRCULAR QUARTIC. 
12. The differentials da, dy can be expressed in terms of a single differential da, viz. 
sm co 
y— ~~r b an d 
then we have 
V/ + § 3 Vy+0 
»=(/+«)(?+«). 
dx =- $*** d y=% xd °- 
It is to be observed that when the dirigent conic is an ellipse, a is a real angle, and 
0 is positive (whence also ^/0 is real and positive), but when the dirigent conic is a 
hyperbola, a is imaginary, and 0 is negative ; we have, however, in either case 
and we may therefore write 
dco 
ds 
V© \/(f+6)W + (ff+0 JV’ 
where s /(f+0yx 2 -\-(g-\-6fy 2 is positive; ds is the increment of arc on the conic 
this arc being measured in a determinate sense, and therefore ds 
being positive or negative as the case may be : has thus a real positive or negative 
value, even when a is imaginary, and it is convenient to retain it in the formulae. 
13. It may further be noticed that if v denote the inclination to the axis of x of the 
tangent to the dirigent conic at the point y/f-\-6 cos a, sin a (v is Casey’s 0), then 
y=^ where V=(f+6) cos 2 v + (g + 0) sin 2 «, 
viz. we have 
cos co cos v sin co sin 0 
V/+0 - u ’ \ / fTd~ u ’ 
giving, as is easily verified, we have therefore 
dco do 
(cc* + y*) V© "(^ 2 +*/ 2 
v© ; 
= dv, 
~^=(x 2 +y 2 )dv, 
dco 
which is another interpretation of -^==. 
14. Substituting for dx, dy their values, the formulae become 
dY =7l{ (/+«>+^5(-(?+9)yX+(/+^Y)}&. 
