1892-93.] Lord McLaren on Elimination of Sines and Cosines. 145 
Elimination of Powers of Sines and Cosines between 
Two Equations. By the Hon. Lord M‘Laren. 
(Read July 17, 1893.) 
This paper is an extension to higher powers of the process which 
I applied to the elimination of 0 from the equations of the Ellipse- 
Glissette {Proc. Roy. Soc. Edin., xix. p. 89). It depends on the 
principle that where the highest power or powers of one of the 
variables are awanting, derived equations can always he formed by 
eliminating a power of each of the other variables between the two 
primitives. 
Where the quantities to be eliminated are the sine and cosine of 
a variable angle, if we put x for sin 9, y for cos 0, and intro- 
duce a homogenic quantity we have always a third equation, 
x^-\-y^ -?? = 0 and by means of this relation we may depress 
either x, y, or 2 in the primitives to the 1st power. 
I shall suppose this preliminary operation to be performed and 
the equations to be given in the form 
+ a^x'^~‘^yz + ayx"^~^z^ + a^x"^~^y^ -H , &c. = 0 . 
(l i.e., 0 = {a^x -1- a^j)xJ^-^ -1- {a^x -i- a^y)x^~h + {a^x + a^y)x^'h^ + - + {a^x a^y -I- 
Similarly, 
(2 0 = {h^x + h^)x^~^ -h (f^x -1- h^y)x^-h + {h^x + h^ifx^-h^ + - + (h-^x -h -t- 
These equations consist of 2?^-f 1 terms — ^.e., they are of the ordei\ 
2n -1- 1 — when written in the 1st of the preceding forms. Consider- 
ing the equations, as written in the 2nd form, it is evident that by 
following the method of Bezout, we may eliminate powers of cc and 
in n-\ different ways, and thus obtain n- \ independent equa- 
tions of the 2nd dimension in the coefficients, as thus : — 
From these pairs we form the n-\ derived equations {of the 
degree), LM' - L'M = 0, NP' - NT = 0, &c. 
In the equations (1), (2), above written, the quantity ?/ has been 
reduced to the 1st degree by substitution from (3). But we may 
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