114 PEOF. A. E. H. LOVE ON THE TRANSMISSION OF 



In like manner the most important part of 



IS 



k'a n + l 



2t (k'a) n+l 

 Thus we have the exact result that, as ma tends to become infinite, 



k'a n+l \}r n (k'a) 3a 



tends to i as a limit. Hence the second factor of the right-hand member of (30) may 

 be replaced approximately by 



Fi {+%(*)} 



The right-hand member of (29) then takes the form of two series, of which the first 

 is that which would occur alone if there were no resistance, and the second represents 

 the correction for resistance. 



8. The 'solution in the form of a series for the effect of curvature without resistance 

 was found as early as 1903 by MACDOXALD( I ). He proceeded to sum the series 

 approximately by substituting approximate values for the BESSEL'S functions which 

 occur in the definition of the functions \[r n and E B . The amount of diffraction which 

 he thus found was so great that his result was challenged by Lord R,AYLEIGH( 2 ) and 

 PoiNCARE( 3 ). In particular Lord RAYLEIGH pointed out that the terms of the series 

 which would contribute most to the result would be those for which n is large of the 

 order ka. MACDONALD(') at once admitted the justice of the criticisms, and revised 

 his calculations so far as to give an independent proof that the amount of diffraction 

 at a considerable angular distance from the originating doublet must be very small. 

 He held then, and has always maintained, that the effect of resistance would amount 

 to no more than a practically unimportant correction. 



9. With a view to the approximate summation of the series it is convenient to 

 introduce the notation 



( )" S" + 1 E n (z) = \/BW e ~ t4 " = V n~i U n> 



(-)'*" + V (*) = v/R. sin fc, = u u L (33) 





If we identify z with ka, we may write R Bi0 , 0, i0 , for the corresponding functions 

 of kr . It is easy to prove the identities 



