Mr. J. Cockle on the Method of Symmetric Products. 493 



2. Any n symbols v^, v^, . , «„ may be considered as the roots 

 of an equation of the form 



v'' + SjV"-^ + s^v''-^ + . . + s,^ = 0. 



Let V, be a linear and homogeneous function of these symbols 

 and of the form indicated by 



Also let the product of m such functions be denoted by tt {v ), 

 so that 



Then, if m=n—l and n be not greater than 4, we may so deter- 

 mine a, 13, ... as to render the product Tr^^.ily^) a symmetric 

 function of v. The case in which n = 2 is scarcely an exception, 

 for we have 



and the anomalous function corresponding to the symmetric pro- 

 duct is, in a manner, symmetric. I shall denote this last func- 

 tion by P'. 



3. The m factors constituting a product may be regarded as 

 the roots of 



Y"' + t,Y"'-' + t.,y'"-'+..+t^=0. 

 If when the products are symmetric we make • 



the values of V which constitute the above two symmetric pro- 

 ducts are derived from the respective equations 



V3_(4z;i + *j)V2 + (8V + -ls,?;,-s,2 + 4s2)V-P3=0j 

 in both of which cases 



ti = nvi + Si, 



and /, and t^ are functions in which r, is the only symbol that 

 occui's unsymmetrically. 



4. Let the result of the elimination of x, between 



a" +pix»- ' +^2^»-2 + . . +p„=0, 

 and 



be represented by 



Then, when n is either 3 or 4, / is so constructed as to make 

 "■"-iW or P„ vanish. Wc arc thus conducted to solutions of 

 cubic and biquadratic equations. It is a sign of the gcucra,lity 



