1896.] Elastic Solid Shells of nearly Spherical form. 63 



When the surfaces are no longer spherical, but are given by 

 (1) and (2), the solution (3) is obviously incomplete, and we 

 require the addition of some terms from the general solution 

 contained in my paper above referred to in Vol. XV. of the Trans- 

 actions. For brevity I shall simply refer by number to the 

 equations of that paper, distinguishing them by the letter A. 

 The terms required are the "mixed radial and transverse" of 

 degree i containing the harmonic o-^. For these from (30,,), (31^), 

 (32^) we have 



U = CT,; 



( , , m- 2n 



2 (2i + 3) n 



+ r i-i Z .-r- i (i + 1)m+2n Y ■ +r-^Z-' 

 + l ' 2(2i-l)n *— 1 + 7 z -^ 



d+1 (i + 3) ro + 2n 

 2(2t + 3)(* + l)n * 



1 • i rr ,. (t — 2) m — 2-n TT . 1 ._„ „ 



i 2(2t-l)iw i+ 1 * 



do-i 

 v =d6 



•(5), 



•(6), 



w = —. — -z j-7 l [same expression as inside square bracket in (6)] 



CD- 



Here it should be noticed Y iy Z it Y-i- 1} Z_i_ x are simply 

 arbitrary constant multipliers of the harmonic ctj, and not them- 

 selves surface harmonics as in the paper from which the results are 

 quoted. They are determined presently from the surface condi- 

 tions. These constants are all of the order e of small quantities ; 

 the terms containing them will thus be called " subsidiary terms " 

 when it is desired to distinguish them from the " principal terms " 

 as given by (3). 



The contributions of the subsidiary terms to the stresses rr, 



r&, r<\> are given in (33^), (34,), (35^) ; their contributions to the 

 other stresses are not required for our present purpose. 



As terms of order e 2 are neglected, the surface conditions over 

 (1) take the form 



--> da; -~i 1 d(Ti ""■; /ox 



rr-e-j^rd- e - — aUT r< P = ~P (°)> 



do sin 6 d<p T 



d6 sin 6 d(f> r dv r 



^> da~i<>. 1 da-i -r^ 1 dai ... nx 



Y dd r sm 6 d(f> rr sin d<f> r 



