132 Proceedirigs of Royal Society of Edinhurgh. [sess. 
Expanding and regrouping, tlie expression is easily transformed 
to 
l±3\ 
2t, ) 
= 0 
or 
16 
-e!-el-el + 2e^e^e,f = 0 . 
The factor in brackets is the square of the determinant 
^1 
1 
and thus Dr Muir’s result is reproduced when we insert the values 
of gg) % ill terms of A, B, C, A', B', C'. 
One interesting point of the transformation seems to be the 
breaking up of this determinant into the four factors above speci- 
fied ; so that the equation 
1 (sin 20) ^ 
(sin 20)~^ 1 
(sin 2a)-' (sin 2/3)"' 
(sin 2a) ' 
(sin 2/3) 
1 
has for roots, as values of tan 0, 
1 
tan a tan ^ , 
tan a 
and 
tan p 
tan a tan p ' tan fS ’ tan a 
But the novelty and value of the process seem to lie in the mode 
in which the elimination is effected by mere general reasoning. 
