082 
DE. J. CASEY ON CYCLXDES AND SPHEEO-QUAETI CS. 
249. From the equation, art. 247, 
(1-j -Jc 1 — 2k cos § f sinr=F cos ^ — cos F, 
which we may put in the form 
(l+7r — 2/r cos sin r=C, 
where C is the small circle cutting J orthogonally and concentric with F, we have th is 
other equation, 
(k—e? 4-i){k-e ~* v - 1 ) sinV=C* (160) 
Hence the imaginary lines k—e s k — e~ ss/ ~ 1 are tangents to the sphero-Cartesian. 
Hence the centre of J is a focus ; and similarly the centres of J', J" are foci. 
250. The equation 
(l-|-7; 2 — ( lk cos g) sin 2 r— C 2 
is the envelope of the circle 
1 + Jc 1 — 2 Jc cos sin 2 r — 0 ; 
but 
C—k cos c> — (cos 5 cos & — sin g sin & cos 0 ) = 0 , 
.*. l+F 2 -f-y/7r cos ^+|M - 2 sin 2 r=(2k-\-(jj cos c$) cos q — ft sin S sin g cos 0 . 
Now the equation of a circle (see art. 36) is 
cos It = cos n cos g-j-sin n sin g cos 0. 
Multiplying by an indeterminate constant, we get 
Hence 
X cos cos ^+i u ' 2 sin 2 r, 
X cos n=2k-\-(jj cos c$, 
X sin n= — [Jj sin &. 
t-, 1 A lr -f a/c cos S + a 2 sin 2 r 
COS XV rrrrr — : 
\//x 2 + 4/x.Ar cos 8 + 4/c 2 
(161) 
Now if It be equal to zero, the circle whose radii is It will be a focus, since it will have 
imaginary double contact with the sphero-Cartesian ; but 11=0 gives the biquadratic in 
(1 -\-k 2 -\-\jjk cos ci -f - [>? sin 2 r) 2 =([jj--\-i[jJc cos ^ + 47 -2 ), . . (162) 
showing that there are four foci, as we know otherwise, since there are three single foci 
and one triple focus. 
251. If E= q , we have the equation 
\-\-k 2 -\-(/jk cos ^-f-^ 2 sin 2 r=0, (163) 
a quadratic showing that a sphero-Cartesian has two great circles which have double 
contact with it. These are not, however, the only great circles which have double 
contact with the sphero-Cartesian. These correspond to the great circle passing 
