1048 HON. LORD M'LAREN ON SYSTEMS OF 



extension to any number of variable quantities, was communicated to me by Dr Thomas 

 Muir after reading the first sketch of this paper : — 



Since x = a, y = b is manifestly a solution of the equation 



A x n + A l x n ~ 1 y + . . . +A n y n = \ci"+A 1 a n - 1 b+ . . . + A n b„, 



=P say, 

 then 



_ a _ b 



%— t> y~~ 



Vn Vn 



is a solution of the equation 



A x n +A 1 x n ~ 1 y+ . . . +A n y n = l. 



For, substituting afp*, b/p* for x, y, the left-hand side becomes 



. a", . a n ~ x b , , . b n . p ., 



A . -+A 1 + . . . +A„— ; %.e., ^=1 . 



p p P P 



This proof, as well as that formerly given, is applicable to functions of any number 

 of variables. For example, the equation 



has the algebraic solution 



x i +y i + 6y 3 z+7xyz 2 =l , 



a 

 x = 



y= 



z = 



^a i +b i +6b 3 c + 7abc 2 

 b 



c 

 2ja i +b i +6b s c + 7abc' i 



And quite generally we can formulate as follows : — 



If <f> be a homogeneous function of the nth degree in r variables, the equation 



has the algebraic solution 



r\\ j 1^/9 J wo j • • • j vC"rj — -L 



<JC-\ — 



V^( tt l > «2 > • • • » a r) 

 Og 



£/0(a! ,a 2 ,...,a r ) 

 a, 



VCn — 



JOo 



3~ n 



X r = 



V#»i , a 2 , . . . , Or) 



%/4>(a lt a 2 , . . ., a r ) 



