1897 - 98 .] Mr Nanson on the Pllipse-Glissette Problem. 159 
so that c is the radius of the director circle of the disc and d, r are 
the distances of the tracing point from the centre of the disc and 
from the intersection of the guides. 
We have also the equation 
(1,1, - 1, 0, 0, 0) (cos 6 , sin 6, l) 2 = 0 . . (4) 
and. the result of eliminating cos 0, sin 6 from (2), (3), (4) is 
4AA' = <h 2 . . . . (5) 
where A, A' are the bordered discriminants of (3), (4), and $ is 
the intermediate of A, A'. Thus we have 
A H G l 
H B F m 
G F C n 
l m n 
A' = l 2 + m 2 - n 2 
<h = (B - C)Z 2 + (A - C)m 2 - ( A + B )n 2 
+ 2¥mn + 2Gnl-2Hlm. 
Substituting the values of the coefficients we find 
A = (A, 4 + 4 p 2 q 2 ){P + (c 2 - cZ 2 ) 2 } 
- 16 (a 2 — b 2 )xy(px + qy)(qx —py) 
-2 {c 2 -d 2 ){(a 2 -b 2 ) 2 -d±}r 2 
A' = - [r 2 - (c - cZ) 2 } { r 2 - (c + d) 2 } 
<f> = 8(a 2 q 2 + b 2 p 2 ){x 2 - y 2 ) - 1 6(a 2 - b 2 )pqxy . 
Thus the curve A = 0 is a quartic which reduces to four straight 
lines through the origin when c = d, that is, when the tracing 
point is on the director circle of the disc. The curve A' = 0 con- 
sists of two concentric circles with common centre at the origin, 
and having for radii the maximum and minimum distances c + cZ, 
c ru d of the tracing point from the origin. The curve $ = 0 con- 
sists of two perpendicular straight lines through the origin. The 
form of (5) shows that the glissette touches the quartic A and 
the circles A' at the points where these curves are met by the 
lines <3>. 
