428 ON THE CALCULATION OF THE [54 



Substituting in the above equation, and writing for simplicity C r instead 

 of C r n , as we may do without ambiguity, we have 



- C, -C,-&c.-C n + n-% = 0. 



Now by Staudt's Theorem the fraction ^ occurs in each of the fractions 

 f n ; hence the quantity arising from this fraction in (?,/, + C.,f, + &c. + C n f n 

 will be 



Also, by the same Theorem, if 2r+l=p be an odd prime number, the 

 fraction - will occur in each of the fractions f r> /, f sr , &c. 



Hence the part of <?,/+ C,/, + &c. which contains - will be 



Also C n = 2n + 1 ; hence by substitution and transposition, we find 

 ( - 1 )-' (2n + 1 ) J n = - {CJ, + (7/3 + &c.} + {CJ, + CJ t + &c.} 



, 



i(c a 



i(a 



^-(C 

 &c. 



2r3r 



+ &c., 

 which gives / when / I. l ...I n _ l are known. 



