2C, ME E. CUNNINGHAM ON THE NOKMAL SERIES 



which, since the coefficients of 8 p - 2 do not vanish, at once give the values of P _/ +1 ... ; 

 these, as in the case of P ^ when the o-'s were unequal, can be shown, if the Toots T^... 

 undergo a permutation, to undergo the same permutation, so that the same P , 2 is 

 associated with any particular T in whatever place this T is taken. 



If, however, the roots T fall into groups of which the members are equal to one 

 another, these equations again resolve themselves into equations of condition owing 

 to the vanishing of ^^(TJ), &c. ; and, as before, the quantities p - 2 fall into corre- 

 sponding groups given as the roots of equations of degrees equal to the numbers in 

 the respective groups. 



The process may clearly be carried on as far as the determination of 2 by the use 

 of the operators A,,_ 2 ...A 2 . 



A further remark should be made as to the finding of lt in connection with the 

 operator A 1( which has been defined to be such that 



Suppose that 2 ...6 2 are given by a set of equations 



a*^ 1 ) = 0, ^(di) = 0, ..., Wl (0/) = 0, 



where the affixes of the w's denote the degrees of the equations, and the roots of each 

 equation are the remaining roots of the preceding after any one of them has been 

 chosen. 



Suppose that of these 2 ...#/ are equal, so that 



Then 6,^(0;) = 0, o,*_ a (0 a 8 ) = 0, ..., w ,_ A+] (0/-i) = 0, but <o,_ A (0/) =/= 0. 

 Then if the X equations following u> k be denoted successively by 



i* = 0, 2 w,t.= 0, ..., 



i(a k is independent of x^, &c., by virtue of &>* = and the preceding equations, and 

 therefore 



Aid*,) = *_(#) 0, since 0J = P >, 0^ = 0^, ..., 



so that ,o) A is also independent of 6? and must therefore vanish identically when a l is 

 determined. Hence also 2 o> k is independent of*! 1 , &c., and therefore 



