of Sound by Spheroids and Disks. 365 



obtain more comprehensive results for obstacles of other 

 than spherical shape. 



The differential equation to be solved is of common 

 occurrence in physical theory, and two important solutions 

 ha\ T e been given. Niven* has discussed the equation in 

 detail in a general manner, but his treatment is not well 

 adapted to problems of the class here contemplated, which 

 admit of a comparatively simple analysis. 



The same remark applies to an investigation by Maclaurinf, 

 which has little in common with that of the present paper. 



Maclaurin's method is peculiarly appropriate to problems 

 in which periods of a vibrating system are sought, but, like 

 that of Niven, is unsuited to diffraction problems concerning 

 small obstacles, mainly because the exact correspondence 

 between solutions valid near the obstacle and at a great dis- 

 tance is left undetermined to the extent of an arbitrary 

 constant. In other words, the asymptotic expansions, at a 

 great distance, of the functions must be definitely known. 

 The method of operators used by Lord Rayleigh in the case 

 of the Bessel functions is here applied to a determination of 

 these expansions. 



The initial reduction of the equation o£ wave-motion 



(V 2 + /r)^ = (1) 



proceeds on the usual lines. 



Treating the case in which the spheroid is ovary, and 

 writing 



A' = pcos<£, y — p sin <£, 



z-±-tp = c cosh (« + i/3), (2) 



or z = c cosh a cos /3, p = c sinh a sin /5, 



where (a, /3) are ellipsoidal coordinates in the plane of any 

 section (denned by <£) through the axis, the equation (1) 

 becomes 



|!± + |?£ + co th«5* + cot/5||+ (coseclr. + cosec^)|^ 



oa- a/3- 0« OP d$ 2 



-f o> 2 (cosh 2 a. — cos 2 j3)-^r — 0, 

 where a> denotes he. 



In the symmetrical case in which waves impinge along the 

 direction of the axis, ~dj~d<^ = 0, and 



t=2A(*)B(/3), . 



* Phil. Trans. 1880, p. 138. f Camb. Phil. Trans. 1903. 



P/iil. Mag. S. 6. Yol. 14. No. 81. Sept. 1907, 2 C 



