432 
PROFESSOR J. CASEY ON A NEW 
Cor. 3. OK/ — OK=diameter of generating circle of the bicircular quartic; for, 
denoting the angle K'OP'=KOP by 4/, we have 
OK' cos \{/— OK cos \|/=OP'— OP=2CP cos 4/ ; 
.'. OK — OK =2 CP = diameter of generating circle. 
Section II . — Rectification of Bicircular Quart ics. 
131. If through the point O (see diagram, art. 130) we draw a consecutive line 
OQQ', then the perpendiculars to this line at the points Q, Q' will pass through the 
points K, K, and we have evidently 
P'Q'-PQ 
dtp 
=OK — OK. 
Hence, denoting the elements P'Q', PQ of the quartic by ds' and ds , we have 
ds'-ds 
= 2 ? 
(273) 
if g denotes the radius CP of the generating circle (see Cor. 3, art. 130). 
132. Mr. W. Roberts showed, in Liouville’s Journal, vol. xv. p. 194, “Sur les arcs 
des Lignes Aplanetiques,” that the difference between two arcs of a Cartesian oval is 
expressed by an arc of an ellipse ; and Professor Genocchi showed some time afterwards, 
in Tortolini’s c Annali,’ that the arc of a Cartesian oval is the sum of three elliptic arcs. 
We propose in this section to extend these theorems to bicirculars in general. We will 
show that Genocchi’s theorem is an immediate inference from Roberts’s, and that each 
is only a particular case of a more general theorem which holds for all bicirculars, and 
which can be expressed in terms of the radii of the generating circles of these curves. 
In order that we may not have to be referring to other writings, we shall investigate 
briefly the leading properties of these curves, referring for a fuller discussion to the 
author’s memoir on Bicirculars. 
133. In art. 128, equation (270), we have the polar equation of a bicircular quartic. 
This, expressed in Cartesian coordinates, is 
4 (d V ■ + ) = {x 1 + y 2 + 2/a- -f- 2 gy-\-r 2 ) 2 (274) 
This equation is the envelope of the conic 
S+^C-f^ 2 , (275) 
where S represents the expression aV-f- % 2 , and C the circle x 2 -\-y 2 -{-2fx-{-2gy-\-r' 2 =0. 
Now the discriminant of the equation (275) is 
„ 2„2 
=arW, 
(276) 
a biquadratic equation showing that there are four pairs of double tangents. 
If the four values of p be denoted by p 2 , ^ 3 , ^ 4 , we have the equations of the four 
pairs of double tangents to the bicircular (see my memoir “ On Bicirculars,” art. 47). 
These pairs of lines are 
S+^,C+^ 2 , S-f-/M.jC +JH, 2 , &c. ; 
