DR. J. CASEY ON CYCLIDES AND SPIIERO-QUARTICS. 
717 
349. If the cyclides W, W', W "pass through a common curve , the cyclides W, W', W'( 
are inscribed in the same cyclic developable (see art. 335). 
Demonstration. The focal quadrics of W, W', W'{ are 
>2 „2 u 2 J2 
a + b ^ c W* 
X 2 , ft 2 
F^-,+t+L+f 
a o' c 
2 
d” 
f; 
a"A 3 &V 
^ "B 
ace 
cd 
d"f 
dd 1 
Now if W /, =W + /i:W' 5 we must have a"=a-{-7ca' 8c c. Hence F'^F+^F', that is, 
the focal quadrics pass through a common curve. Hence the proposition is proved. 
350. If the cyclide W" pass through the curve (WW 1 ), the cyclide W" is inscribed in 
six cyclides passing through (W W'). 
Demonstration. If two cyclides, Q, O', be such that one is inscribed in the other, then 
the reciprocal of O with respect to 12' is inscribed in O', and also the reciprocal of 12' 
with respect to O. 
Let O =A cd -f- B$ 2 + Cy 2 + Dfr 2 , O' = A'« 2 -j- B'j3 2 -f- C'y 2 -f- D^ 2 . Hence the required con- 
dition will be the determinant 
A, 
B, 
C, 
D, 
A', 
B', 
C', 
H, 
A 2 
B 2 
c 2 
D 2 
A'’ 
B M 
O’ 
W’ 
A ' 2 
B ' 2 
C ' 2 
D ' 2 
A’ 
~C> 
T) ’ 
or aBCD " A 7 B , L 7 I)'( A 6, — A, B)( AG' — A'C)( AD' — A D)(BC' — B'C)(CD' — C'D)(BD' — B'D) = 
Hence ( AB' - A'B)(AC'— A'C)(AD' - AD)(BC'' - B'C)(CD' - C'D)(BD' - B'D) = 0 (232) 
is the condition of O being inscribed in O', that is, the product of the six determinants 
of the matrix 
A, 
B, 
c, 
B, 
A, 
B', 
G, 
i y. 
as is otherwise evident. 
To apply the condition (232), we have 
''' lb' cd 
XXJII aa ' 2 | 03 | CC ' 2 | ^ >2 A . 
1 — a-\-ka' b + kb c + kd^ ~^~d + kd'° ’ 
and let the cyclide through the curve (W W') be 
(a-{-h l a')K 2 -\-(b-\-h J b l )ld 2 k'c')f-\-(d-P Id d')b 2 =0, 
then we get, after some reduction, the condition of W, being inscribed in W -j-/dW' 
5 E 2 
