96 
Proceedings of Royal Society of Edinhurgh. [sess. 
This gives a determinant of the 3rd order with (1), (2), and (9), and 
the reduced eliminant takes the form 
(B2 + A2)(x^ + y^) + 2(B2 - ~ B^(2 A + 4C)(^r2 + ^2) + 4B2C(A + C) = 0 . 
The curve consists of four ovals. When the guides are rectangular 
we have 
a = |; A = 0; B = (a2-&2): c = i(^^2 + 52 ). 
and the equation reduces to 
{a? - + y^ + 2xY) - 2{a^ - b^ia^ + b^^^ + y^-) -{- (a^ - + b^ = 0 ; 
or, 
{{Y + y^) - {a? + &^)}^ = 0 . 
The ovals are then attenuated to two coincident circular arcs, which 
is the known form of the locus of the centre of an ellipse moving 
between rectangular guides. 
Since this paper was in type, I have found that an equation 
equivalent to the equation (6) of the determinant given on p. 89 
may be formed by a method identical in principle with Bezont’s 
system of elimination for homogeneous equations. The original 
generalised equations may be stated thus — 
{(C-A)X + BY + aJ. X+f^SgY + y^ + AHO I . . . (2) 
{ -/iX + AY}.X+{vY + ;a}=0 j , , . (3)-Y 
•• { B,. - /3.A} Y2 + {(C - A)./ + /32/>*}XY + {y, + C}/xX + + Byi.} Y + a,_ix - X{y^ + A) 
The determinant may be arranged as under, where E is put for 
A- 
C: 
Y-2 
XY 
X 
Y 
(1) 
(c) 
A 
n 
V 
(1) 
(a) 
E 
B 
a., 
^■2 
y, + C 
(2) 
(c).Y 
A 
V 
= 0 (3) 
(e)x 
( 
-A) 
n 
V 
A 
(4) 
(d) 
By 
- /3f, - ] 
hX, (y^ + C)/x, 
av 4- B/x, 
ag/A - A(y2 + A) 
(5) 
This determinant may be reduced to the 4th order by combining 
(2) successively with (3) (4) and (5), so as to cause the 1st column 
to vanish ; and as the coefficient of Y^ in (2) only contains the con- 
stant E (or A - C), this can be done without raising the degree of the 
eliminant. The eliminant contains eighteen compound terms, and 
when developed is, of course, identical with that previously found, 
as I have verified by a partial expansion. 
