of a Beam of Heterogeneous Rontgen Rays. 783 



Writing equation (1) in the form 



x n —piX n ~ l - s rp 2 x n ~ 2 — . . . = 0, 

 since the roots are f, rj, £, . . . , the polynomial is equal to 



(x-£)(x-r,)(x-Z) 



Hence 



= x n —p x x n ~ x -\-p 2 x n ~ 2 — p 3 .r"~ 3 , 

 and equating coefficients we get 



Si + f =p u 



5 2 + £Si. = p 2 , 



5 3 + £S 2 — P3, 



f S w _i = p n . 



Thus all the S's can be found in terms of f . 

 Differentiating the identity 



x n - Pl x n - l +p 2 x n ~ 2 = (x-%)(z- v )(x-£) . . . , 

 and putting x=£ after differentiation we find 



n% n - l -(n-l)p 1 % n - 2 -h(n-2)p 2 Z n - 3 -... 



= (f-i)ff-0(f-f).... 



Thus both the numerator and denominator of (2) can be 

 expressed in terms of f without knowing tf 9 J, . . . . 



If two roots are equal, say f and 77, the expressions for 

 both A x and A 2 become indefinite. A 1 + A 2 , the quantity 

 we require, has, however, the definite value 



_ yogn-2— yi?*-3 + y 2 7„- 4 • . 



where 



?i + f = Si, 



73 + ??2 = S 3, 



where the S's are the same as in equation (3). The de- 

 nominator is 



(„_l)^-2_( yi _2)S ] p- 3 +(»-3)S 2 p- 4 ... 



Mr. R. A. Herman of Trinity College, Cambridge, has 

 pointed out to me that the determinant in equation (2) is of 

 the type discussed by Sylvester in a paper in the Philosophical 

 Magazine, Nov. 1851, and called bv him a canonizant. 



