Integration in the Problem of Diffraction. 375 



problem, where the boundary is formed by two planes 

 inclosing an angle a. As a special case his formula yields 

 the solution given by Sommerfeld* for the semi-infinite 

 plane. 



The method by means of which Macdonald obtains the 

 solution of this problem depends upon a general theorem, 

 proved earlier in his treatise. I propose in this note to give 

 another demonstration, using the method of contour in- 

 tegrals. This method has been employed with very great 

 success in the theory of potential, by Dougallf, and the 

 proof which I give here would occur to any one reading his 

 most important and suggestive paper (cf. §§ 17, 21). 



Taking the origin in the edge of the wedge, which occupies 

 the space a<#<27r, and considering the case in which the 

 electric force is parallel to the edge of the wedge, the problem 

 reduces to the solution of the equation 



where, in addition, it is required that u shall vanish at the 

 boundary — {) and 6 = a, and that it shall become infinite 



as log ( ^ J, when K = 0, at the point (V, 0'). 



tided we 



where 



R 2 =r 2 .-rr' 2 -2rr' cos(0-6'), 



and U n (z) is the Bessel's function of the second kind, given 

 by the equation 



u -«-*iES?( J -*«-'- < -J-W> 



The addition formula for U (£R) is well known, and will be 

 used later, namely 



U (*R)=J (Ar')Uo(*r) +2% l=1 J n {kr , )V n [kr) cos n{6-6 f ) 



* Sommerfeld, " Mathenmtische Theorie der Diffraction,'' Math. Ann. 

 Bd. xlvii. (1896) ; cf also a paper by the Author in Proc. Loud. Math, 

 Soc. vol. XXX. 



f Dougall, " The Determination of Green's Function bv means of 

 Cylindrical or Spherical Harmonics/' Proc. Edin. Math. Soc. vol. xviii. 

 (1900). 



2 C 2 



If the space were unbounded we should take as our solu- 

 tion 



