140 PROF. H. M. MACDONALD ON THE DIFFRACTION OF 



The principal part of the first of these integrals arises from values of in the 

 neighbourhood of = p. l/ -'e /3 ^, and of the second from values of in the neighbourhood 

 of = fL k e' l!a "-, hence writing in the first = p'W*" + 1} and in the second 

 = /A 1/2 e" 1/3ir ' + i, the expression becomes 



le-' > f e -*''- VV'^fU' (%!-&'* f e-V'-V".-'"-"*.-*' ^ 



J -n'V'"' J -n'''-e '"" 



and the principal part of this expression is equal to the principal part of 



which, writing , = r)e~ ]>/> '" in the first, and , = ije 1 '' 6 " in the second, is equal to 



i-*,* 1 '" 3 ,-x <!-''>' 



l e -' >' | e-** "-* /v efy-ie 1 '*" e~* '*-* " v dr, ; 



- M ' 3 t J/i /2 t 



that is, equal to 



now the principal parts of the first and second integrals in this expression are equal, 

 but with opposite signs, and therefore the principal part is 



Hence 

 and 



tan ^ = ^e- 4 *' 1 ' 1 ......... (viii.). 



The leading terms having been determined, the approximation can be carried further 

 by using the differential equation, and the result is 



Ac. 



1.5 _ 3 1.5.7.11 _, 



* It should be observed that the constant of the imaginary part is half the value that would have been 

 given by STOKES' rule, p. 112 of the paper referred to above; the explanation of this is that the value of 

 the 6 in STOKES' investigation that corresponds to this solution is one that belongs to a boundary for the 

 intervals of 6 ; this case is not discussed by STOKES, but it is not difficult to prove that for such values the 

 constant takes half the value it has in the interval. 



