1891 - 92 .] Hon. Lord M'Laren on the Ellipse-Glissette. 
91 
This equation has been developed in two different combinations, 
and the results compared and found to agree. 
Instead of forming the determinant of the 5th order from that of 
the 8th order, as above given, we may form the 5th row directly 
from the rows marked (2) and (4), as follows : — 
By eliminating successively the 1st and 2nd terms between these 
rows, we form the two equations 
(B/x + CA)XY . +(Bv+ aX)X + X/3Y + {j3+ y)X = 0 
(B/x + CA) Y2 + (a/x - Cv)X + PfxY + (y/x - CA) = 0 . 
By subtraction, — after multiplying the first equation by Y and the 
second by X, — we obtain the 5th row, being (6) of the tabular series. 
The operation is essentially the same as that already performed. 
The determinant may be reduced to the 4th order by eliminating 
one of the higher terms between the 1st row and each of the others 
successively. But this is really a retrograde step ; because in the 
development, the number of terms is greater than in the preceding. 
Apparently, to obtain the best results, the order of the determinant 
ought not to he less than the number corresponding to the number 
of terms of the original equation. 
The equations of the glissette, as given by Professor Tait {Proc. 
Roy. Soc. Edin.^ vol. xvii. p. 2), are 
cc - ?’cos(^ - a) = (a^cos^^ + S^sin^^)^ .... [a) 
y - rsin(^ - «) = (6^cos^^ + a%in2^)= . ... (p) 
By expansion, and putting r cos a =p, r sin a = (-q), and expressing 
cos^^ in terms of sin^^, these equations become 
(cd - + 2^)siiY(9+ 2pqc,os,0s>m0 2px QOS, 6 - 2qxs\n0 -yA - _ q 
(a^ - + q^)mdO - '2pqQos,6Bi\\6 + 2gycos0 + 2pyBm6-^h^ — q^ = 0 
Also 
2{px + qy) qobB + 2{y)y - qx) sin 0 + -py^ - (f - x"^ - y^ = 0 \ 
Comparing these questions with the generalised expressions (1), 
(2), and (3), we have 
B =cd — h^ —^2 ^ ^2 . 
“2 = ^iy ; 
A. =%px + qy ) ; 
C = -2m; 
p = 2{py - <2x) ; 
V =a^ + h^ -p^^ - qp-x- - y“. 
