368 Dr. J. W. Nicholson on the Scattering 



Thus, quoting the results, 



a) 2 2ft) 4 



4ft) 6 



Xo_ 3 "135 



+ 8505 + ••• 





6o) 4 4ft) 6 



"875 "65,625 



X 2 =,6+ n 



94cw 4 



+ 9261 + •• 



23 



X 3 = 12+^ft) 2 + ... 

 4o 



X 4 = 20 + ||^+..., (14) 



in agreement with the results o£ Niven's general calculation. 

 The first two harmonics become 



H (,)=l-^ + ^M±V + (15) 



H^-^.^ + ... . (16, 



From the differential equation, it may be shown in the 

 usual manner that 



{n — m . n + m -f 1 + or . e ;i - e m + ft) 4 S„ — S M + . . . } J 1 H n H TO rf/x- 



= (T^ s .H n H w '-H w H B ')i 1 . 



The quantities e, S,... being algebraic in m and w, and 

 H ffl , H 3i and their derivates finite at fi? = l 9 it follows that 

 when m and ?i are unequal 



j-H„ i H,^=0. ..... (17) 



-1 



gain, C 1 TT o , J^ of- 28&) 4 \ 



J_ l H »^ = 2 ( 1 ~« + 2025 + -) 

 f 1 2/ 3» 2 267o>" 



while to the second order, 



j>* =!(>+«) <"> 



For all important applications H 3 (/u) may be taken as 



K 



