26 
Proceedings of Royal Society of Edinburgh. [sess. 
there results the quadratic in cos Q 
cos^ 6 + 4(A + B - r"^){jpx + qy) cos b 
+ (A + B-r^)^- 4(g'X-j:>3/)2 = 0 .... (a). 
Again, if in (1) and (2) the terms involving the first power of 
sin 0 be alone retained on the left-hand side, sin B may he eliminated 
between the two equations by division, and thus a second quadratic 
in cos^ obtained, viz.. 
{(A - B + ^p^)qx-\- (B - A-}- 2^2)j9y}cos 
-1- { (A -I- ~K)yq -pqr^^ - 2{p^ -1- (f)xy]o,o^ 0 
+ {qjx^^y -f qxy^^ — Aqx — hpy] =0 (/?). 
This, it may be noted, is the equation also arrived at by using (S) 
to eliminate sin0 from either (1) or (2). 
Writing now the equations (a) and (^S), for shortness’ sake, in 
the forms 
L2COS ^0 + M3COS 0 -f = 0| 
X^cos - 1 - 6 + V 3 = 0 J , 
where the suffixes indicate the degrees of the coefficients as regards 
X and y, the resultant desired is known to be 
• L, 
L2 Mg 
. X, 
Xi ft 2 
or 
^2/^2 “ X^Mg 
L2 vg - Xj^isr^ 
Mg N, 
H ^3 
= 0 , 
Xl^4 
M3I/3 - 
= 0 
This equation is seen to be of the degree, and the difficulty — 
not by any means a small one — has to be attacked of finding the 
extraneous quadratic factor. After many trials a fortunate observa- 
tion led to the discovery that the factor is 
{qx-pyf, 
and then a lengthy and inelegant transformation brought about its 
ejection and the final equation 
