1046 



HON. LORD M'LAREN ON SYSTEMS OF 



This method of finding solutions of indeterminate quantities is hereafter referred to as 

 the method of Homogeneous Variation, because all the quantities are varied proportion- 

 ately in order to obtain a new series of values. 



Although only two variables, x and y, are here expressed, the explanation of the 

 method of solution is intended to cover the case of an equation of three or more variables. 

 In order to simplify the illustrations as much as possible, I shall generally suppose two 

 variable quantities x and y ; or r cos 6 and r sin 6 ; w is then the parameter or inter- 

 cept on the axis of X. It is convenient to take this quantity = unity, which can always 

 be done by dividing out. 



In order that the series of points to be found may correspond to equal angular 



intervals, it is best to assume & , . and 7} x . , equal respectively to the cosine and sine 



of an angle. Then x 1 __ j/i... are proportional to the same cosine and sine, and are 



the rectangular coordinates of the curve to the argument $. 



Example 1. 



x i + 2x s y + 3x 2 y 2 + 4>xy 3 + 2y i = w i =l. 



For the sake of clearness, I shall, in this example only, dispense with the use of 

 logarithmic tables, and find two values of x and y from auxiliary equations in which the 

 assumed values are whole numbers. 



(l) Let £i = l; 171 = 1. The sum of the terms of the auxiliary equation is 12; 



.\w x = 12 1/4 ; x l = g/wi = ^ = y x . 



This may be verified as follows: — Let x x and y x have the values here found. Then 

 taking the terms of the equation in their order, 



c4 ~\12W ~12' 2a;32/ ~ 2 (l2w(l2W - 12' 



and so on ; and the terms are as under- 



X* 



2x 3 y 



3x 2 y 2 



4xy 3 



2y* 



1 



12 



2 

 12 



3 

 12 



4 

 12 



2 



12 



Sum of the terms = =~ = 1 > as ** should be. 



(2) Let £ 2 = 1 5 Vz= 2 ; the sum of the terms of the auxiliary equation is 



1+4 + 12 + 32 + 32 = 81; .-. w 2 = 81 1 /* = 3; x 2 = ^w 2 = ^; y 2 = v /w. 2 = ^ 



This solution may be verified in the same way as the preceding without the use of 



logarithms. 



