1890 - 91 .] 
Prof. Tait on Dr Sang’s Paper. 
339 
The elimination of tan <f> between (1) and (2) is easily effected 
by multiplying (2) twice over by tan</>, using (1) after each opera- 
tion. We thus avoid the radicals which make Dr Sang’s work so 
complicated, and we have only to eliminate tan <£ and (tan <£) 2 
among three equations of the first degree. The resulting equation is 
of the fourth degree in (sin v) 2 , but it contains the irrelevant factor 
(cos v) 2 (sin v ) 2 
+ 
p 2 
[Another method of effecting the elimination, while quite as simple 
as that just given, has the advantage of not introducing the irrelevant 
factor. Write for shortness 
cos V 
=p 
sin v 
and we have 
p 
= 2 
p cos <jf> + q sin tj> = — , 
y?(sin </>) 3 + q( cos <£) 3 = 0 . 
From the second of these, by the help of the first, we at once obtain 
p sin (f> + q cos <£ = — cos </> sin 
\y 
l 
or 
V + 2 
cos <f> sin <f> y ' 
The following are immediate consequences : — obtained, respectively, 
by multiplying together the first and fourth of these equations, and 
by squaring and adding the first and third : — 
p 2 + q 2 + a y 
pq 
sin cos y 2 
p 2 + q 2 + ipq sin <£ cos </> = -^/l + (sin cf> cos <£) 2 
y \ 
From these the final result may be written by inspection, in the form 
4 p 2 q 2 1 
p A + q A + 
=- 2 i + 
p 2 q 2 
( v 2 + <F - ^ - ±pY(p 2 + - ^ 2 ) 
p 2 q 2 
-,2 5 
which is obviously of the third degree in (sin v) 2 . 
