DE. J. CASEY OX CYCLIDES AKD SPHERO-QUAETICS. 
703 
313. Denoting the discriminants of W, W' by A, A', we have A, A' given by the 
determinants 
a, 
n, 
m. 
ib 
a' , 
ri, 
m', 
y, 
A ~ 
n, 
h 
l 
, A = 
n ! , 
v, 
V , 
m, 
l, 
c , 
r, 
m', 
l', 
c' , 
F, 
2, 
r , 
d, 
y , 
q', 
F, 
d'. 
Then the discriminant of FW+W' will be got from A by writing in place of a, b, &c. 
ka-\-a', 7cb-\-b\ &c. ; the result will be a quartic in 7c, which I shall call the invariant 
equation of the two cyclides, and write in the form 
£ 4 A+£ 3 0+& 2 O+£0'-f A'=0 (189) 
Since there are four values of 7c which satisfy this equation, we see that through the 
curve (WW'j can be drawn four binodal cyclides, that is, four cyclides each having two 
conic nodes. If we eliminate 7c between FW+W' and (189), we shall get the equation 
of these four binodal cyclides, namely, 
A'W 4 — 0'W 3 W' + <hW 2 W' 2 — 0WW' 3 + AW ,4 = 0. . . . (190) 
314. Since the equations of W, W' are the same in form as the tangential equation 
of their focal quadrics F, F', and if F, F' touch, W, W' will have double contact, hence 
it follows that the condition of W, W' having double contact is the vanishing of the 
discriminant of the invariant equation (189); .\ the tact-invariant of W, W' is 
4(12AA' — 300' + <I> 2 ) 3 — •(72AA'<I>-f-900'{I>— 27A0' 2 — 27A'0 2 — 2d> 3 ) 2 . . (191) 
315. The tact-invariants of two conics and two quadrics are the analytical expression 
of remarkable geometrical properties which have not been hitherto noticed by any writer 
so far as I am aware ; on this account, and because extensions of them hold for the tact- 
invariants of two bicircular quartics and two cyclides, I shall give their investigations here, 
and we shall incidentally find results that are important independently of the properties 
that w T e have alluded to, and which we now proceed to demonstrate. 
316. If A, B, C, D be four points ranged in alternate order on a right line, the six 
anharmonic ratios of A, B, C, D can be expressed ^ 
in a way that bears a remarkable analogy to the 
six trigonometrical functions of an angle. 
On A B and C D describe circles ; let O, O' be 
their centres, P one of their points of intersection, 
then the angle O P O' equal angle of intersection 
of the circles ; and taking the six anharmonic 
ratios of A, B, C, D, as given in Townsend’s 
‘ Modern Geometry,’ or Ciiasles’s ‘ Geometrie 
Superieure,’ it is easy to see, if we denote the angle O P O' by 0, that we shall have the 
equations : 
