142 Mr. Murphy on the 



Let us retake for a moment the same equation 



(ss - a) m = k F {%), 



if we extract the m ih root of both sides and suppose 1, p, P 1 ... P m -' 

 to be the »« th roots of unity, we may form m different equations, 

 viz. 



= a + h m .F(%y 



1 1 



= a + P .h m .F(s)' 



1 1 



% = a + p 2 .A m .i^(s)" 



*'= a + P m - 1 .h'".F(z) m 



to each of these equations apply Lagrange's Theorem to determine 

 f(z), we thus get m values, viz. 



/(*) =/(«) + If ./' (a) . F («)"■ + — . \f («) . Fa- }' + &c. 



2 



/(*) =/(«) + P A"/' («) . F(ai» +^~.\f (a) . F~W"\' + &c. 



n .//"' 



■ &c. 



,m =/(«) + P " - ■. h-.f (a) . f (a)- + > j i 2 " */'(«) F(«rr+ 



Adding up all these equations, and recollecting by the proper- 

 ties of the roots of unity that their sum, or the sum of any of 

 their powers is 0, except in the case of the m lh , 2m*, &c. powers, 

 when the sum is m, we evidently get for the sum of the function 



