ELECTRIC WAVES ROUND A PERFECTLY REFLECTING OBSTACLE. 127 



sum should be of the order 0(^1, ^ir i must be small, that is, the point P must 

 be on or close to the boundary of the sphere described on OOi as diameter. In 

 these terms of the series S 3 and S 4 , e 2l< * 0+Xo) can be replaced by 1, and then the 

 sum of the terms of the series 81 S 3 + S 3 S 4 required is the same as the sum of the 

 corresponding terms of the series 



-u-. 



' dp 



Now, for the values of 6 specified above, -j- { < fa (n + %)6} can only be small 



when a is a right angle or differs from a right angle by a small quantity, hence the 

 principal part of the sum of the terms of the series Si S 2 + S 3 S 4 is the same as the 

 principal part of i/^, that is, it is equal to i^. 



G. When r > i\, (j> and (^ are interchanged, the value of i/; may be written 



,/, = S',-S' 3 + S 3 -S 4 , 



and the discussion of the series is identical with that of the previous case. If a plane 

 be drawn through d perpendicular to OOj, then for points P on the side of this 



plane remote from the sphere and not close to the plane the principal part of S' 2 is ^i ; 

 for points outside -the geometrical shadow and not close to its boundary the terms 

 of the series S 4 , for which n is less than z , have a sum of the same order as i/> 1; which 

 represents the effect of reflected waves ; for points outside the geometrical shadow 

 between the plane through O^ and the sphere centre O, radius O0 1; the terms of the 



series S 4 , for which n is greater than z u , have a sum whose principal part is \lt ly 

 and for points in the neighbourhood of the plane through d the terms of i/;, for 



which \n + ^Zi is of lower order than z/ 73 , have a sum whose principal part is i//j. 

 Thus for all points outside the geometrical shadow and not close to its boundary the 

 principal part of the value of i/> obtained from the series is the same as that which 

 would be given by applying the methods of geometrical optics. 



7. The effect of the terms of the series S 3 and S 4 , beginning with those for which 

 z n% is of the same or of lower order than z^ 1 , has now to be obtained. The 

 corresponding part of S 4 may be written 



S 43 = (2ir)- v (sin 0)'/ (/c^ 2 )- 1 2 (n + )'/' R'/,R i 'V[ ! *, + ^.-*-*,-(+', s )+'/>] > 



--0-A-V/3 



now for values of n greater than z it follows, from the corresponding expression 

 for 2< + 2x [Appendix (vi.)], that the factor e s ' ( * 1+x ' )> does not oscillate but approaches 

 the value 1 as n increases, hence for these values of n it may be written 



