106 CARl'MAEL ON THE DP]TERMINATION IN TKEMS OF A 



will be rapidly convergent when n is large. The value of 'p {x, in. ?i) may accordiugly 

 be Ibuiid approximately from (xi), for small values of ot, if we calcu.late the A^alues of — 



/.' COS (j- ^ — 7t) _1 



H 



for a few values of m. 



Ill order to obtaiu the value of this integral, let us first take the well-knowu integral 



J ' ''y = -z 



n 



For 1/ write -/.i . ^' we find that 





n 



Integrating both sides of this equation with respect to x, p times between the limits 

 and X we find 



-f «-" 1.3 (2p - 1) - ^ .>■ 



or, making x = ^ 



2 

 ft 



•' H-p ^ ^^ 1.3 {2p—l)'^ 2 



Again take the well-known integral 



— a- ff- V n : 



f cos2rHdH= 2^ £ rt- 







/J 



and in it write - for ^, and put à- = § we obtain 



ft' a .. 



f a ^ cosr H .dH=y-^ ,i ^ 

 n 



Integrate with respect to /• between the limits and r we get 



^ cos (/■ H — — ) 



/, 6 - .d.H^y^:;fe -^^ dr (xiii) 



" ' 



Integrate again with respect to r between the limits and r we have 



ft' ens (r ft — 2^) ,■ H ^ c , - ii ;•- -Ir 1 



