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PEOFESSOE J. CASEY ON A NEW 
142. If the bicircular becomes a Cartesian oval, the point C will be at infinity, and 
we shall have (see cor. 1, art. 140), 
ds ! "-ds=0. 
Hence 
fd&"= 0, 
and dd= s dQ+g'dQ'+p"d&'; (296) 
.*. s'=$ s dQ+$dd +j YdQ" (297) 
This is Genocchi’s theorem. 
Cor. By integrating the equation 
ds , -ds=2 i dd 
we get 
s'_ s =2fedQ, 
(298) 
which is Roberts’s theorem. 
143. To reduce the integral §gdQ to the normal form of elliptic integrals. 
X 1 y* 
Let the focal conic of the bicircular be 1=0, and the corresponding circle 
of inversion J=(#— f) 2J r{y — </) 2 — >£ 2 =0. The equation of a tangent to the conic is 
x cos Q-\-y sin Q=\/ a 2 cos 2 0-fi£ 2 sin 2 0, 
or say 
x cos b-\-y sin S=ja ; 
therefore if x' y' be the point of contact, we have 
x 1 cos 6 
y' sin 6 
¥ y 
Now if <p be the eccentric angle, x’ = a cos <p, y'= b sin <p ; 
cos cos 6 
a p 5 
Hence 
sin <p sin $ _ 
b p ’ 
tan Q=- tan <z>. 
a r 
d&~- 
abd(f> 
' a 2 sin 2 <p + Z» 2 cos 2 p 
(299) 
Again, since g is the radius of the generating circle whose centre is the point (x y') 
on the local conic F, and which cuts the circle J orthogonally, we have 
