DR. J. CASEY OX CYCLIDES AND SPHERO-QTJARTICS. 
653 
the touched circle 
x-+if+yx +2f'y +c' -0, (1) 
the orthogonal circle 
a 2 + y'x + 2f‘ "y -f- c" = 0 . (2) 
The given conditions supply the two equations, 
Hf+r-o)(f+f n -C)=( 2 S9 '+ 2 ff-e-o'r, 
2<w"+2#"-c-c"=0. 
Hence, eliminating c, and putting a’, y in place of — g, — f\ which are the coordinates 
of the centre of the variable circle, we get for the required locus 
%' s +/ B -«'X* , +y 5 +2^'+2/"2/+< ; ")={20'-?>+2(/'-/>+«'-c'r, (US) 
a conic which has double contact with the circle cut orthogonally, the radical axis of 
the two fixed circles being the chord of contact. 
The focus of the conic is the centre of the fixed circle ; this is most easily seen by 
taking the centre of the fixed circle as origin ; then^^O, g' = 0, and the equation (118) 
becomes that of a conic having the focus as origin, namely 
4:c'(x 2 +y 2 ) + (2g"x-\-2f"y+c'+c") 2 =0 (119) 
Now if the circle (1) be an osculating circle of a bicircular quartic, and the circle (2) 
one of its circles of inversion J, the conic (119) must have three consecutive points 
common with the focal conic of the quartic which corresponds to J, namely the centres 
of the three generating circles of the quartic which the circle (1) touches. Hence we 
see that the proposition is proved, that the evolute of a bicircular quartic is the locus of 
the foci of a variable conic which has double contact with a circle of inversion of the 
quartic , and which osculates the corresponding focal conic. 
174. The theorem proved in the last article enables us to determine the degree of the 
evolute of a bicircular. For let v be Chasles’s characteristic ; that is, let v be the number 
of conics osculating the focal conic F of a bicircular quartic, and having double contact 
with the corresponding circle of inversion J, which can be described to touch a given 
line ; then the required degree will be ov . Hence the degree of the evolute will be 
known when v is found. We shall prove in the next article that v is 12 ; therefore the 
degree is 36 ; but this number sutlers a reduction, as we shall prove that it includes the 
fine at infinity taken 24 times. Hence the reduced degree is 12. 
175. To find Chasles’s characteristic v for a system of conics osculating one given conic 
and having double contact with another given conic. Our solution will depend, 1°, on 
the question, If a variable conic touch a given fine, and have double contact with a given 
fixed conic, to find the envelope of its chord of contact with the fixed conic. 
This is solved as follows. The condition that the conic S-J-P" 2 should touch P' is the 
tact-invariant 
(1 + S")S'-R 2 =0 (see art. 46). 
Let S=# 2 -| -y 2 -fz 2 , ¥" = ’kx-\-u,y J r vz, V=X'x-\-fy-{-dz, and the tact-invariant gives 
4 u 2 
