444 
PROFESSOR CATLET ON THE BICIRCULAR QUARTIC. 
positive. In reference to a figure which I constructed I found it convenient to take 
0 3 , 0 O , Q 2 to be in order of increasing magnitude : this being so we ha vef-\-0 3 positive, 
g-\-6 3 negative ; and the other like quantities g-\-Q x , g-\-6 w g-\-^ all 
positive : we then have y\ and <y\ each positive, y 2 0 negative, y\ positive : viz. the conics 
and circles are 
Hyperbola H 3 corresponding to real circle C 3 , 
Ellipse Ej „ real circle C 15 
„ E 0 „ imaginary circle C 0 , 
(viz. the radius is a pure imaginary) 
„ E 2 „ real circle C 2 , 
and where the confocal ellipses E ls E 0 , E 2 are in order of increasing magnitude. The 
centre C 0 is here a point within the triangle formed by the remaining three centres 
C 1? C 2 , C 3 . It will be convenient to adopt throughout the foregoing determination as 
to reality. 
7. It may be remarked that a circle of a pure imaginary radius y, =?‘X, where X is 
real, may be indicated by means of the concentric circle radius X, which is the concentric 
orthotomic circle ; and that a circle which cuts at right angles the original circle cuts 
diametrally (that is, at the extremities of a diameter) the substituted circle radius X ; 
we have thus a real construction in relation to a circle of inversion of pure imaginary 
radius. 
Investigation of dS. — Art. Nos. 8 to 17. 
X 2 Y 2 
8. The coordinates of a point on the dirigent conic may be taken to 
be {f-\-6)x, ( g-\-Q)y : ana we hence prove as follows the fundamental theorem for the 
generation of the bicircular quartic. Consider the generating circle, centre ( f-\-6)x, 
(g-\-Q)y, which cuts at right angles the circle of inversion (X— a) 2 + (Y— |3) 2 =y 2 . If for 
a moment the radius is called c5, then the equation of the generating circle is 
(X-/+9^+(Y-FTW=y; 
the condition for the intersection at right angles is 
( a _/+fc)’+( ( 3 -<?T #)■=/+»■, 
and hence eliminating ci 2 , the equation of the generating circle is 
X*+Y>-i-2(X-«X/+<>-2(Y-|3)(?+%=0; 
and considering herein x, y as variable parameters connected by the foregoing equation 
(/+ 6)x 2 -\-(g-\- % 2 = 1 , we have as the envelope of this circle the required bicircular 
quartic. 
9. It is convenient to write R=^(X 2 +Y 2 — Jc ) ; the equation then is 
E-(X-a)(/+D>-(Y-/3)( ? + %=0 ; 
