MISCELLANEOUS PROPOSITIONS. 409 



Let the n + 1 values be denoted by a^, a? 2 ,... x n _ 1 , besides 

 h and h. We shall shew that any one of the former n I 

 roots of (1) occurs twice in (1). For the derived equation 

 of (1) is 



=0 ..................... (3); 



and any one of the values x iy # 2 ,... #_, is by supposition a 

 root of the equation <f>'(x) =0, and so satisfies (3). 



Hence we have by the Theory of Equations 



But by supposition the roots of the equation <j> (x~) = are 

 a? lf C 2 ,... x^\ hence 



<f>'(x} = n(x-x 1 } (oj-ag... (a? -,_,); 

 therefore fofc)] 2 -^ ^'^ ^~^ .......... ..(4). 



Differentiate (4) with respect to a;; thus we get 



ri*<j> (x) = x<j> (x) + (x* -K>) <}>" (x] ............ (5). 



From (5) by equating the coefficients of x n , x"' 1 , a;"" 2 , ... we 

 shall be able to determine in succession p lt P 9 >p a f> Foi 

 thus we have , 



n 2 = n + n (n 1), 



n * pi =(n-l) Pl + (n - 1) (n - 2) Pl > 

 n*P* = ( n ~ 2) JP 2 + (n - 2) (n - 3) p a - n (n 



n *P*r ( n ^ 4 )ip4 + (n - 4) (n - 5) p 4 - (n - 2) (n - 

 and so on. 



';>ri A n ^ A 



'Thus ^ = 0, ^ 2 = - , _p 8 =0, ^ 4 = -. 



Therefore 



^ n(n-4)(n-5) 



~ 



8 



