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1044 HON. LORD M'LAREN ON SYSTEMS OF 



1. Principle of Homogeneous or Linear Variation of a Function. 



If F(a, /3, y . . .)" = be a homogeneous function of the n th degree of any number of 

 quantities, a, /3, y, . . . ; and if a lt j3 u y u &c., be known values of these quantities satisfy- 

 ing the equation, then may another set of values, a 2 , /3 2 , yz, satisfying the equation, be 

 found by multiplying or dividing each term by any desired factor, m n . 



Let the function consist of a series of homogeneous terms of powers of the quantities 

 a, y8, y, . . . multiplied by coefficients *p x q 1 ... and equated to zero. Let a v fi v y x be 

 values satisfying the equation ; which accordingly will be of the form 



«"+p 1 a 1 »- 1 /3 1 +p 2 a 1 »- 1 /3 2 + . . . q P 1 n + q 1 p 1 n - 1 y 1 + q 2 p 1 n - 2 y!+ . . . + r j .<-*- ? P?7i' ? + ... =0; 



where the last term is the generalised term for three quantities. 



To find a new series of values satisfying the equation, we have only to multiply every 

 term by the same numerical quantity, m n . The equation is, of course, unaltered in value, 

 and is now of the form 



i 1 ) n +2\(ma 1 ) n -\ml3 1 )+p 2 (ma 1 ) n - 2 (m/3) 2 + . . . g ( m A) B + 9i( TO P\)" _1 (w7i) 

 + r,.(ma 1 ) B -*-»(m i 8 1 y , (my 1 )»+ ... -0, 



where the term in the second line is the generalised form of a term resulting from the 

 multiplication of the function by m n . 



By writing a 2 for ma,, /3 2 for mfi v y 2 for my v &c, the equation is restored to its 

 original form, with a new set of values, a 2 , /3 2 , y 2 , of these indeterminate quantities satisfy- 

 ing the equation. Comparing the two sets of values, we find the relation 



« 2 £ 2 72 

 «i Pi 7i 



which was to be proved. 



The preceding proof evidently includes the cases of negative, reciprocal, and fractional 

 indices. 



In the preceding theorem it is not assumed that all the quantities a, /?, y, . . . 

 are variables ; and the proof is evidently the same, whether all the quantities are con- 

 ceived as being subject to indefinite variation, or whether some of them are conceived as 

 having only certain definite values from which values of the other quantities are to be 

 obtained. For example, if a 1 j3 1 are variable coordinates, and y : is a parameter, the set 

 of values a v /3 : , y x represents a point on a plane curve of the nth degree having the para- 

 meter y x , and the set of values a 2 , /3 2 , y 2 represents a corresponding point on a similar 

 curve whose parameter is y 2 . But if the three quantities a, )8, y are all conceived as being 

 subject to indefinite variation, y being then a third coordinate, the function represents a 

 conical surface of the nth. degree, and the two sets of values then represent corresponding 

 points on parallel, and therefore similar, plane sections of this surface. 



Again, certain of the quantities may represent the coordinates of a point on a central 



