ELECTRIC WAVES ROUND A PERFECTLY REFLECTING OBSTACLE. 121 



Substituting for <, x<>> &-> their approximate values, this requires that 2 KI + 

 is small, since & is small, and this is impossible for %ir > > , ?TT > > i ; hence 

 the sum of these terms is negligible in comparison with x/^. The terms of the series 

 for which |z w | is' of lower order than z v ' are of the order (/ca) 3/2 (/c^)" 2 1/ 1; and 



their sum is of highest order when j-(2<f> +2x<) < (f>injr) is small, or nearly equal 

 to a multiple of 2?r, for a value of n in this series. In this case the approximation 

 for ( u + Xo is I 7r + - 3~ l/ * (3p.) 2 [Appendix (vi.)], and the above condition becomes that 



* 



ira.a. l ir is small, or nearly a multiple of 2-rr, which is satisfied if both a and a t 

 are small. The sum of these terms then is 



(, . .... _. .._ .. i ?5_z_z.4.I/.6lJ-l/.12(a''2"''ff-l;-"'3_^l_,.-H 



3 sin ff cos 



. 



retaining only the most important part, and the sum is therefore of the order 

 (/ca) 3/1+1/3 (/cr 1 )~ 2 x/ 1 , which is of lower order than t/ 1; and therefore negligible in 

 comparison with it. When n + ^ is greater than 2 and n + ^z is of the same or 

 of higher order than z 1/3 , but n+^ is less than z and zn^ is of the same or 

 of higher order than z' /3 , when z<z 1; the terms are of the order (/cr) l ' 2 (/cr 1 )~ 2 i/; 1 , and 



their sum will be of the same order as i/^, if -= (2<^ + 2^ < ^ I + WTT) is small, or 



nearly a multiple of 2?r, for a value of n in this series. This condition is satisfied if 

 there is a value of n in the series for which IT a O. I TT is small, or nearly a multiple 

 of 2ir, which requires that both a and a t should be small. For the other values of n, 

 which have to be taken account of, the order of the terms is (itr)'' 1 (ieTi)~ a i^, and the 

 condition that their sum should be of the same order as \fj 1 is, that for some of these 

 values of n, ^rra^n- is small, which is impossible. Hence the part of the series S 2 

 which is of the order of x/^ arises from the terms beginning with a term for which n 

 is equal to z + A2 1/3 , where A is a positive quantity of the order of unity, and the 

 principal part of S 3 is therefore equal to the principal part of 



- L sin ff cos \ff 2 (w + ^R'/.R^VW.**.-*-*.-"-) Jj {(2n+ 1) sin W'} 



**CJ"i 1,,+AV 3 



sin ff cos 10' 2 



Now 



that is [Appendix (x.)] 



where T is a negative quantity and T increases rapidly with n. Also, when r 



VOL. COX. A. R 



