1891-92.] Hon. Lord McLaren on the Ellipse-Glissette. 
95 
The coefficients in [a) (b) (c) (d) (e) constitute the required 
determinant of the 5th order, and the eliminant is of the 8th degree. 
The eliminant of three general equations of the 2nd degree in x, 
y, may be formed in like manner : only in this case the coefficients 
are all of the 3rd degree, and are of the determinant form 
h< 
^1 1 
1 ^ 1 ’ 
d, 1 
(Xg, 
^2’ 
^2 
«25 
^2’ 
^35 
^3 1 
1 «3’ 
1 
Two equations may be formed in the form (6) of the glissette 
problem, and two others by eliminating x and dividing by y, and 
conversely. Any three of these together with the three original 
equations constitute a determinant of the 6th order ; and the 
eliminant is of the 12th degree. I have worked out the general 
solution, but it is not necessary to give it here as the method has 
been fully explained. 
As an example of the application of this method, I take the 
case of the locus of the centre of a generating ellipse moving in 
contact with guides inclined at any angle, a. The equations 
(original and reduced) are 
a^GOS^(<fi - a) + 5%in^(<^ - a) = x^ (1) ( 
a2cos^(^ + a) + 5%in2(<^ + a) = (2) | 
- 52j(cos2a.cos^(^ + sin2a.sin(^cos<^) + ((x2sin2a + 5^cos^«) -x^=0 (1) I 
{a^ - 52)(cos2a.cos2<j5> - sin 2a. sin cos <^) + (a%in^a + b^cos^a) -y^ = 0 (2) j 
Putting A and B for the coefficients of cos^^ and smcf>cos(f) respec- 
tively, X for cos<j6, and Y for sin^, and multiplying by and XY 
successively, we have the following scheme, which is of the 7th 
order, but is reducible to the 3rd : — 
X4 X3Y X2Y2 XY3 X2 XY 1 
A 
B 
C — X^ 
(1) 
• 
• 
A 
-B 
c-y^^ 
(2) 
A B 
• 
C-^2 
(3) 
A -B 
C-lf 
. 
= 0: (4) 
1 
1 
-1 
. 
(6) 
A 
B 
e — Qe^ 
(6) 
A 
-B 
(7) 
1 
1 
-1 
(8) 
From (3, 4, 5) 
-2A 
• 2(A + C)-*2-J,2 
From (6, 7, 8) 
B 
?/2 - 
Whence, 
• 2B(A + C)-B(k2_2/2) 
(M 
1 
(M 
• 
(9) 
