ICO Dr. J. W. Nicholson on the Bending of 



Again, 2 S iumnK m (z) ei m 'l7r=J. a (z)-e m 'J m (z), 



leading on reduction, m being half an odd integer, to 



K ro (*) =2-' (6/*)* (/;+/,-#) e-i"*\ . (73) 



Comparing with the usual substitution in terms of R n and 

 <f) n) it follows that 



B % =(27r)- 1 6fW(/ 1 H/ 4 2 )\ , 



^=tan~ 1 / 1 // 4 J' " ' • ^ 



where / 4 stands for/ 2 -f/ 3 . If the functions F correspond to 

 the functions/ with argument p 1 — (m i — kr)(6/kr)$, then if 



tan <£ wr = F ly /F 4 , "j 



R, ir =(27r)- 1 6§r 1 -*(F 1 2 -fF 4 2 )./- " ' l ° j 



If an accent applied to any function denote differentiation 

 with respect to its argument p or p x as the case may be, 



^=Br/a#-(6/#)*^ s -(6/*,)*^-^ 



since r -1 'dpfdx = 'dp/'dm=(6/z)i. 



Thus /6\i/-4/V-/./7 /6y F,lV-F,F/ 



which may be reduced to a simpler form as follows : — 



The integral equivalent to u=f 2 — if 1 has been shown by 

 Stokes to satisfy 



where the accent means 'd/'bp. Thus separating real and 

 imaginary elements, 



/•"-fc/i— j. 



A sain, it is not difficult to show that 



and by addition, since /,=/ 2 +/ 3 



so that / 4 and/i satisfy the same equation, from which 



/.A"-/i//=o, 



and therefore Jifi — fiU is independent of p. 



