226 
ON THE BICIRCULAR QUARTIC. 
[667 
5. The equation of the bicircular quartic may be expressed in the four forms 
(X 2 + F 2 - k ) 2 — 4 [(/ + 0 ) (X - a ) 2 + (g + 0 ) ( Y - (3 ) 2 ] = 0, 
(X 2 + F 2 - W -4 [(/+ 0 1 ) (X - a,) 2 + (g + 0 1 ){Y— A) 2 ] = 0, 
(X 2 + F 2 - hY - 4 [(/+ 0 2 ) (X - a 2 ) 2 + (g + 0 2 ) (F- /3 2 ) 2 ] = 0, 
(X 2 + F 2 - k 3 y - 4 [(/ + 0 3 ) (X - ar 3 ) 2 + (gr + 0 3 ) ( F - &) 2 ] = 0, 
the equivalence of which is easily verified by means of the foregoing relations. 
Determination as to Reality. Art. Nos. 6 and 7. 
6. To fix the ideas, suppose that f— g is positive; then in order that the centres 
of the four circles of inversion may be real, we must have f+0.f+0i*f+0a-f+0s 
positive, but g + 0. g + 0!. g + 0 2 . g + 0 :t negative ; and this will be the case if f+ 0, 
f+01, f+0-2, f+03 are all positive, but g + 0, g+0 u <f+0>, 9 + 03 one °f them 
negative, and the other three positive. In reference to a figure which I constructed, 
I found it convenient to take 0 3 , 0 lf 0 a , 0. 2 to be in order of increasing magnitude: 
this being so, we have /+ 0 3 positive, g + 0 3 negative; and the other like quantities 
f+01, /+#o, f+02> 9+01, 9+0o. 9+0-2 all positive: we then have y 3 2 and each 
positive, y„ 2 negative, y 2 2 positive: viz. the conics and circles are 
Hyperbola H 3 , corresponding to real circle C 3 , 
Ellipse E x , „ real circle C x , 
„ E 0 , „ imaginary circle G 0 , 
(viz. the radius is a pure imaginary), 
„ E. 2 , „ real circle C 2 , 
and the confocal ellipses E x , E 0 , E 3 are in order of increasing magnitude. The 
centre Co is here a point within the triangle formed by the remaining three centres 
C\, C. 2 , G 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 7, =i\, 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. 
8. The coordinates of a point on the dirigent conic 
Investigation of dS. Art. Nos. 8 to 17. 
X 2 F 2 
= 1 may be 
(g+0)y: and we hence prove as follows the fundamental 
taken to be (/+ 0) x