568 ON POLYZOMAL CURVES. [414 
so that this condition being satisfied for one of the points in question, it will be 
satisfied for the other of them. Now for the three conics 
cz 2 + 2/ yz + 2g zx +2h xy = 0, 
c'z 2 +2f'yz + 2g'zx 4- 2 Jixy = 0, 
c"z- + 2f”yz 4- 2 g”zx 4- 2 li"xy = 0, 
forming the Jacobian, and throwing out the factor s, we may write the equation in 
the form 
Cz 2 + 2Fyz + 2 Gzx + 2Hxy = 0, 
where the values are 
0=g (fc” -f’c’) + g (,f"c -fc”) + g” (fc’ -f'c), 
H = g 0h'f” - h’f) 4- g' (h'f - hf") + g" (hf - h'f), 
2 F=h {fc >f -f’c) + h' {f'c -fc”) + h" (fc -fc), 
2G = h (eg” - c”g) + h' (c”g - eg”) + h” (eg' - c'g); 
and we thence obtain 
cE + hC = - (// -f'g) (c"h - ch") + (fg -fg") (ch! - c'h) 
= 2 (fG+gF), 
viz., the condition is satisfied in regard to the Jacobian and the first of the three 
conics; and it is therefore also satisfied in regard to the Jacobian and the other two 
conics respectively. 
I do not know any general theorem in regard to the Jacobian which gives the 
foregoing theorem of the orthotomic circle. It may be remarked that the use in the 
Memoir of the theorem of the orthotomic circle is not so great as would at first 
sight appear: it fixes the ideas to speak of the orthotomic circle of three given circles 
rather than of their Jacobian, but we are concerned with the orthotomic circle less as 
the circle which cuts at right angles the given circles than as a circle standing in 
a known relation to the given circles. 
Annex II. On Casey’s Theorem for the Circle which touches three given Circles. 
The following two problems are identical: 
1. To find a circle touching three given circles. 
2. To find a cone-sphere (sphere the radius of which is = 0) passing through 
three given points in space. 
In fact, in the first problem if we use z to denote a given constant (which may 
be = 0), then taking a, a’ and i (z — a”) for the coordinates of the centre and for the 
radius of one of the given circles ; and similarly h, b\ i (z — b”); c, c’, i (z — c”) for the