170 Mr. T. P. Kirkman on the General Solution 



the sum of four given values of the ^th power of 



the sum of two given values of the mih power of 

 R z =G 3 Qp 



the sum of three given values of the /th power of Qt, and thus 

 finally to read the function 



the sum of the n values of the asymmetric H5 which we choose 

 to construct on G n . 

 Our theorem is : 



(1) The first sum of values so read, but the last in construc- 

 tion, viz. 



is a given rational and symmetric function of the n variables 

 x i3 x 2) . .. , x n , whatever be the positive exponents /3, y, . . . , b } a,i. 



(2) Every one of the succeeding sums of values, viz. 



is a given irrational and symmetric function of those variables, if 

 the solution of equations of the nth and inferior degrees is known. 



The demonstration is very simple. 



(1) It is evident that 



s m =Rr+,Br=(Qi.+ &fj& m + ( 3 Qi+4Qi+5Qi)'" 



is invariable by T 2 . F 3 , which represents all the six permutations 

 of x lf x 2) x 3 . It follows that 



r 4 s m , or rj si, 



whatever be the integers [a and p, is a function invariable by the 

 114 permutations of x Xi x 2 , x 3 , x A ; whence of necessity 



is invariable by the 115 permutations of x l} x 2 , x 3i x 4 , x 5 ; and so 

 on, till we see that 



is invariable by the permutations of x i} x Q) x 3) . . . , w n ; i. e. the 

 function 2^ is rational and symmetrical in these n variables, 

 whatever be the positive exponents. 



It will be here useful to make a distinction between the roots 

 a i) Oct) - • • ) a n of tne equation 



(x — a 1 ){x — a 2 )(x — a 3 )... (x—a n ^ 1 )(x—a n )=0 = V } 



or the entering roots of U = (as I call them), all asymmetric 



