782 Sir J. J. Thomson on the Absorption Coefficients 

 But if x is equal co an}' o£ the quantities f, 97, £*, &c, 



s _^s 1+l ^s 2 ... =0. 



Hence, eliminating S , S 1? . . , S a from the n-h 1 equations 

 we get 



yo, 2/1, 

 Vi, y>2, 

 &2, y& 



y2, . • 

 ys, .. 

 y* • ■ 



-yn 

 • y»+i 



• yn+2 



y n -\, y n , 



1, X, 



# 2 . . 



y2n-l 



= 0. 



(2) 



Thus the n roots of this equation for x are the values of 



X X G 



f, 77, £ — *\ £., of e l , e 2 . Hence this equation determines 



X X 9 



the values of e 1 , e 2 ; and since 6 is known, the values of 

 the Vs can be immediately determined. 



We can find the values of the coefficients A l5 A 2 , A 3 , &c. 

 as follows : — 



From the first n equations of the system a we get 



3/0, 



1, 



1 



yi» 



V, 



r 



Vn-l, 



v"-\ 



t- 



1, 



1, 



1 



f, 



>?> 



? 



P. 



<? 2 , 



£2 



(3) 



r-\ it 1 - 1 , r- 1 

 (f-l)(f-B(f-«)... 



where S r is the sum of the products of the (ri — 1) quantities 

 97, f, €, . . . taken r at a time. Thus 



S„_i = life, Si = 77 + f+6+ .. . 

 The value of a coefficient can be expressed in terms of the 

 root corresponding to the coefficient, and does not require a 

 knowledge of the other roots. For 77, f, e are the roots of 

 the equation 



x 71 - 1 — Si^ l - 2 +S 2 ^- 3 — . . . = ; 

 so that the polynomial is equal to 



(x-yXx-£)(x-e) 



