291: Dr. J. W. Nicholson on the Bending of 



The derivates of u are only wanted to the first order, and 

 become 



<rx a 



2(1 -x 2 .1-cVj 



<TX~* 



\ 1 + l-x 2+ 1-cVj 



^{3 + 2^ 2 (l + c 2 )-18cV 



" ~ ±(l-x 2 .l-c 2 x 2 )\ 



+ 6cV(l + c 2 )-cV}. 



If in each o£ these we write # = rsin #/R, the value of yp 

 is given by (138) on reduction. But the general result is 

 too complicated to be o£ use, and we shall confine attention 

 to a special case, namely, that in which r/a is large, so that 

 we are investigating the scattering of the waves at a great 

 distance from the sphere. In this case, we may write 

 c = 0, x = sin 0, and deduce 



v 2 = (l-a? 2 )-£ = sec0. 



v 3 = x (l -x 2 ) ** = sec 9 6 tan d. 



u 4 = (l + 2x 2 )(l-x 2 )-$ = (l + 2sin 2 #) sec 5 0. 



v d jv. 2 2 = tan 0. 



^A'2 2 -^' 3 >/ = K3 + sin 2 ^)sec 3 ^ . . . (152) 



uju = i (1 + 2 cos 2 0) sec 2 9 cosec 0. 



uj/tt = i (3 + 2 sin 2 6) sec 4 cosec 2 6. 



Write C = cos 6>, S = sin 6 for brevity. Then by (137) 



it 4VV2 2 3r 2 3 >' f 2 2 ^ ^2^ 



_ 2 

 V 6 v 2 "S 



9-15S 2 + llS 4 



1^C 3 S 2 

 Moreover, by (151), 



(153) 



S 2 __3C_ 1_ , 5 S 2 



I 3 8S 2 80 ' 



9-f2lS 2 +5S 4 



X = 20 3 ~ 8S 2 80 24 3 





24S 2 C 3 > 



and the second approximation to u is 



M = — (— - n Sill 2 #£4 -J 1 + — > 



\a7r cos 6'/ t 2 ) 



(154) 



