ELECTRIC WAVES BOUND A PERFECTLY REFLECTING OBSTACLE. 129 



for n + % = z is of higher order than z ' /3 - Hence the principal part of 843 is equal to 

 the principal part of 



-(27r)-'- (sin 0)'-(/cn 2 )- 1 2 



-0 



when the value of 



for n + ^ = z is of higher order than z ~ 1/3 .* 

 Now 



-(27T)- 1 ' 1 ' (sin ^''"(icr, 8 )- 1 2 



the principal part of the second series has been found above, and arises from the 

 terms on both sides of the term for which 



/i c(b o<z>i 

 C' ~l~ -^ ~l~ 7: - (J . 

 en on 



If H! is the corresponding value of n, and 



^ + 1 = 2 sin a = z t sin a. lt 



! is given by 



e = 7r-a- ai , 



hence, writing n >i, + , the principal part of the above series is 



00 J 00 



Therefore, if D denotes the distance from Oj to the point P (r, 0), 



sin B/D = sin a/?^ = sin a/r, 

 and making the substitution 



a cos a j cos 



* If e is an angle of higher order than o~' /s , y the semivertical angle of the tangent cone to the sphere 

 from Oi, and the tangent cones with vertices on 00! whose semivertical angles are y e are constructed, 

 the space excluded is that between these two tangent cones, which are both close to the boundary of the 

 geometrical shadow. 



VOL. OCX. A. S 



