ELECTRIC WAVES ROUND A PERFECTLY REFLECTING OBSTACLE. 133 



converges to a limit when m increases without limit, the limit S to which it converges 

 will be given approximately* by 



S = J2 



it being understood that only the terms of this series where it converges are to be 

 taken. 



The above clearly fails it a = llmi, for in that case lc a vanishes; it is also 

 inapplicable if a 2kin is very small, for then 1 e" is very small, and the terms of (i.) 

 diverge at once. The result shows that the sum S m , however great m is, is not of an 

 order higher than that of the terms that compose it when a 2km, does not vanish or 

 is not very small ; this may be compared with the known case of DIRICHLET'S integral 

 which vanishes unless the range of integration incloses the origin. 



The second case to be considered is that in which a = (to which the case 

 a = 2&7ri is always reducible) and u n+t contains an exponential factor e m ' ! ; writing 



and remembering that 



where B 2 *_i are BERNOULLI'S numbers, it follows that 



J_ B^, D 2 *- 



ZK i 



When the real part of /3 is not greater than zero the important part of the integral 



em 



e^w n+t dt is that contributed by small values of t, and writing 



w n+ 1 = w n + tw' n + ^w" n + &c. , 

 it follows that 



fm |-m rm 



e^v n+t dt = w n \ e^dt + w' n \ eFtdt+&c.-\ 

 o Jo Jo 



Now 



F<J*dt = ("e^dt- Fddt, 



Jo Jo Jm 



that is 



r^dt = ^(-/5r^+e^/2/3m- &c., 



Jo 



" When w n+t only involves t as a polynomial the series has only a finite number of terms which 

 represent the exact sum. 



t For the determination of the important part of the integral it is sufficient that w n+t should be 

 expressible in powers of / for small values of t. 



