90 Proceedings of Royal Society of Edinlurgh. [ sess . 
The determinant consisting of the first eight rows is a solution, 
but is too high, being, as will presently be seen, of the 9tli degree 
in the coordinates of the curve. The determinant of the 5th order 
consists of the first four rows and the last row, or (2), (3), (4), (5), 
and (6). In applying the dialytic method, as has been done to the 
solution of the present case, I had the choice of nine equations of 
the 3rd degree ; and the selection from these was determined by 
two considerations : — 
1st. In order that the terms of the 3rd degree might be made to 
disappear without unduly raising the degree of the eliminant, it was 
found that the second group of equations should be formed in pairs 
by multiplying two equations of the first group by X and Y respec- 
tively. This is a condition which seems to be generally applicable 
to problems of elimination. 
"2nd. In order that the eliminant should have an absolute term 
(as required by the conditions of the case), it was seen to be neces- 
sary that the constant part of its terms of XY and Y^ should not be 
the same for equation (2) and the resulting equation (6) ; because 
if the constant terms be equal with hke signs, the whole function 
will vanish identically. 
These conditions determine that the second group of equations 
should be formed from (2) and (4) by multiplying these functions 
by X and Y successively. [If we were using (1) or (2) with cos^d 
instead of sin^^, the second group would be formed from (5), which 
would then have the sign of X^ negative.] 
In the preceding table, (/) is formed from (h) and (c), (y) is formed 
from {d) and (e), and (6) is (/) - (y) : the resulting equation (6) 
being only of the 2nd degree. The equations (2), (3), (4), (5), and 
(6) constitute the required determinant of the 5th order. 
An equation conjugate to (6) may be formed by suppressing sin^^ 
in the original equations, and proceeding as above. 
A partial expansion of the quintic determinant gives the follow- 
ing eliminant : — (A) 
{o\ - jS/JL + Bi/) + aXv^ + /3X^fX + Jfjdv — BA^I/ - aX/P - — yX^v] 
-P (fSX + afji - Cv) { CX^v -f ajxv^ + aX^fJL + /SXv'^ — Cv^ — (SX^ — 2yXfiv } 
+ {CX-yfx) { CXfjd -f /3X^V + I3f^v - BX V - B/xi/2 - CXv^ - yX V - } 
-P (B -f y)X . (BX^ + C/xv^ -1- yX^ + ~ BXv^ — CX^/x — a)dv — ajPv} 
•\-Cv- a/x) { 2BX/XF + CX^i/ + aX^/x + a/x^ - C/xY - /3X^ — jSXfjd } — 0 . 
