[569 
ON THE CYCLIDE. 
69 
des to be K x , K 2 and 
, X L 2 may be termed the 
;, or, what is the same 
iay these are the points 
point on the axis are 
,he origin is in this case 
. the vertices there are 
-ough the axis take for 
/ely; and then, according 
he plane at right angles 
else the circles on the 
js according as the two 
neters f, g, h, k assumes 
so that f+g + h + k = 0, 
ose of y, z parallel to 
also 
h) {x — k) = 0, 
0 this equation becomes 
), 
>, 
), 
). 
K 2 are given by 
569] 
and the points L x , L 2 by 
Z- =-(Q-f)(Q-h) = -(Q-k)(Q-g). 
Now writing for a moment 
fi=f+g = -h-k, 
7 =f+h = -k-g, 
8 = /+ k = — g — h, 
we have P — — -J- ~ 
/3 
/38 
2 ~o > Q = — i , and thence PQ = J8 2 . Moreover 
7 
2Y 2 + 2Z 2 + 2(P- Q) 2 
= -(P-f)(P-g)-(P-h)(P-k) 
~(Q ~f)(Q -h)-(Q-k)(Q-g) + 2(P-Q) 2 
= -(fg + hfc +f h + gk) - 4PQ 
= 8 2 - 4>PQ 
= 0, 
Y 2 + Z 2 + (P — Q) 2 = 0, 
which equation expresses that the four points are a system of skew anti-points. 
that is, 
The point x = Q should be a centre of symmetry of the circles G, C'; to verify 
that this is so, transforming to the point in question as origin, the equations are 
2/ 2 + {« + Q -1 (f±g)} 2 ~ i (f-g) 2 = 0, 
y 2 + {x + Q - %(h + &)} 2 —l{k — h) 2 = 0, 
that is, 
y>+j*-i^(S + 7 )j S -i(/-<7)’ = 0, 
+ -!(*-*)» = 0. 
But 8 + y =/- g, 8 — 7 = k — h, so that these equations are 
2/2 + {* ” 21 (/- #)} = i (/- gY, 
y* + I"® ” 2 ^ (k - A)} = Hk - h)\ 
which are of the form 
y 2 + (x— a) 2 = c 2 , 
y 2 + (x — ma) 2 = m 2 c 2 , 
and consequently x — Q is a centre of symmetry of the circles C, G'; and in like 
manner it would appear that x = P is a centre of symmetry of the circles D, D'.