78 
ON THE CYCLIDE. 
[569 
viz. this will be identically true if 
X 2 — X x -f- Ci J 
ß 
V v ßJ' 
X 2 + Zi - Ci = - - X* + 2 c 1 X l 
7 
or, as this last equation may be written 
-|]-a + Cl ’, 
-£,+ 
-*l + 
Z}-c} = Z?~ + 
The equation of the orthotomic sphere is thus found to be 
L - X x - c x /y/(~ ^)} + y 2 + 2 2 - 2*£, + - Ci 2 ^ + 2c 1 X l 
or, what is the same thing, 
a; 2 + y 1 + z 2 — 2zZ, — 2x |x x + Ci j — X x ^ — 2c : . 
or, as this may be written 
“ (- f + f - !) + 2! - ^ ^ " f X ‘ 8 " ^ f) “ 2 ° lXl 
P = 0 ’ 
-|) + c, ! -a = 0, 
viz. this is 
“ris+i - 
1) -H £ 2 — 2mZ % + -I x 
-|) + V. A /( —^1-4=0, 
where Z. 2 is arbitrary. We have thus a series of orthotomic spheres; viz. taking any 
one of these, the envelope of a variable sphere having its centre on the conic 
_ x _|_ X _ i _ o, and cutting at right angles the orthotomic sphere, is a cyclide. The 
P ^ 
centre of the orthotomic sphere is a point at pleasure on the line 
% = X 1 + c 1 ^J ^, y = 0; 
and the sphere passes through the circle z = 0, 
{- x ‘- V(-i)F + » 1+ *‘ -°' a e M k/("?) + \/(-|)} =0 ' 
00^ V 2 
viz. this is a circle having double contact with the conic — — + ^ = 1; or, what is 
the same thing, the orthotomic sphere is a sphere having its centre on the line in 
question, and having double contact with the conic — ^ + — = 1.