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LXXIV. The Scattering of Waves by a Cone. By Professor 

 H. S. Carslaw, The University of Sydney, N.S. W* 



IN view of the interest at present taken in the question of 

 the scattering of waves by a sphere, the corresponding 

 problems for a cone may have some slight value. Recently 

 I have obtained the expression in series which gives the 

 solution for the cone, but 1 have not yet been able to reduce 

 my results to a form suitable for numerical discussion. The 

 method which I follow is similar to that of a former paper 

 on Diffraction f, and is suggested by DougalFs work on 

 Potential J. The proof is hardly suitable for these pages, 

 and I confine myself for the present to a statement of one of 

 the results obtained. 



The vertex of the cone is taken as the origin. Its surface 

 is given by 6 = O , and its axis by = it. 



We start with a source at the point on the axis produced, 

 at a distance r' from the vertex. This is the point (V, 0, 0) 

 in spherical coordinates. 



The disturbance in the infinite medium due to this source 

 is defined by 



e -«eB 



"o 



R ' 



where R 2 = r 2 + r' 2 — 2rr cos 6. 



This can be written 



at 



wo = -4=, 2 ^'(n+iJK i (i«r') J x (*t)P b (a*), 



\/ rr o 2 2 



for r</, with the usual notation for the BessePs Functions §. 



On replacing this series by an equivalent Contour Integral, 



and associating with it the solution required by the surface 



condition u = at 6 = 6 0j we obtain the following result : — 



for r< /, the" summation being for the values of n > — \ which 

 make P„ (fi ) vanish. 



* Communicated "by the Author. 



t Phil. Mag. (6) vol. v. (1903). 



% Proc. Edinburgh Math. Soc. vol. xviii. (1900). 



I Cf. Macdonald, ' Electric Waves/ p. 91. 



