1891-92.] Dr Muir on a ProUem of Elimination. 31 
^4(^2 ^2)4 _ 1 6?^2(^2 _ - 0?){x^ + ?/2 _ ^2 _ 4^2) = Ua%^xhf. 
The question therefore is whether a separate investigation of the 
equation for the case ^ = 0, q — h would confirm or remove the 
suspicion raised in regard to the coefficient of AVhen = 0 
and q — h the original equations become 
- ‘Ibx sin 9 = a‘^ cos^9 } 
y- - 2hy cos 9 = a? sin^ 9 I 
From these we can derive two simple equations in sin 9, cos 9, 
viz. : 
2hx sin 9 - 2hy cos 0 = - P 1 
[a?xV - itP-xy'^) sin 9 + {a^yV - cos ^ = 0 j 
where P stands for x^-^y- - o?. Solving for sin cos 9 we obtain 
r P(a2P - W-lf) \ 2 f P(4&2;^2 _ ^^2p) ^ 2 
I ilyia^Y - 25272") J 1 45^(a2P - Pbh'^) J ^ ’ 
which readily leads to 
^4p4^2 _ 16a2&2a:;2^2p2^p ^ ^2) ^ 40^4p2^2^2^2 _ _ 64<x25^x2^2p^.2 _[_ 045®z2^2r4^ 
so that, on striking out the factor + ^2^ we have the equation in 
the form 
a^P^ - 1652^32^2^^2 _ J,2^p2 _ _ 645%2^2|^2P _ J,2^p 
or 
= - 645^2^2(^2 _ ^2^p + ^ia^^xY , 
or 
^4p4 _ I652.r2y2((x2 - 52)P(P — 452) _ 64a25^ir2y2 j 
and this agrees to the letter with the suspected result obtained by 
specializing from the general equation. 
There thus, I think, can be little doubt that the calculation of 
the coefficients in the general equation has been accurately per- 
formed. 
