1892 - 93 ,] Lord McLaren on Elimination of Sines and Cosines. 147 
{ + a^x^y + a^x‘^z + a^xyz + ayxz^ + a^7jz^ + a^z^ }x+ [a^y -\r a^z}z^ = 0 
{b^x^ + ka.}x-{- [b^y-\-hQz]z^ = 0 
{a^x^ + a^xy + a^xz + a^yz}x'^ + {a^x‘^ + a^^y + a^xz + a^yz + af^}z^ = 0 
{ + &c. + &c. = 0 
{a^x + a 2 y}x^ + {a^x^ + a^xhj + ayx^z + a^xyz + a^jxz^^ + a^ijz^ + a^z^ }z = 0 
{ h^x + l> 2 y}x^ + { h^x^ + &c. }z = 0 
By cross-multiplication each of these pairs furnishes a derived 
equation of the 4th degree in which y"^ is to he replaced by - x^. 
The resulting equations, which follow, are expressed in the usual 
abridged notation where, e.y., the first term of the expression (6), 
f ^1)^3 I is to be read 
{(%^3 ~ %^l) "b (^'^4^2 ~ af^^X^^ , 
and so throughout the paper. 
xhj x^z x^^yz x^z^ xyz? xz? yz? ?}■ 
(4) is to be multiplied by y and (5) is to be multiplied by y only, 
and (6) is to be multiplied by x and y. These five multiples and 
the six multiples of the original equations being arranged in deter- 
minant form as given below, it can be proved by a very simple test 
that the eliminant does not vanish identically. In the diagonal 
from right to left the terms underlined furnish the element afhf 
(by taking the negative part of the expressions from the 4th and 
5th rows, and the positive parts from the 6th, 7th, and 8th rows), 
and it is evident that there is no other such term, because in the 
first and last three lines a-^ and occur once only, and in each of the 
intermediate lines af^ occurs once only. The same proof applies to 
the eliminant of two equations in 6 of the 3rd degree printed in 
the page following, if the lines be arranged in suitable order. 
