68 Dr Ghree, The Equilibrium of certain Shells. [Jan. 27, 



In all cases it will be noticed that for a given magnitude in the 

 maxima values of eai the importance of the subsidiary term in u 

 increases rapidly with i. It has been tacitly assumed that the 

 subsidiary terms are relatively small, so that the solution should 

 not, without further investigation, be applied to cases in which 

 ei 2 <Ti is anywhere comparable to unity. 



The effect of the departure from sphericity on the change of 

 density produced by the pressure is deducible at once from the 

 value of the dilatation A. For the thin shell we have from (10,,) 

 and (101 J, putting in the latter 



Fi = Ri — R'i, 



where R it R{ are given by (16), 



afp - b 3 p Seao-j (p - p') 

 *~ (m-n/3)(a s -b 3 ) (3m-n)h K h 



Thus if p as before be the surface radius vector, 



(a s p - fry) 3(p-a)(p-p' ) 

 (m-n/S)(a s -¥) (3m-n)h v h 



Thence in the case either of internal or external pressure, A is 

 numerically largest — i.e. the change of density of the material is 

 greatest — in the neighbourhood of the longest radii vectors. 



In the case of nearly spherical shells exposed to bodily forces 

 depending on solid harmonics, or surface forces depending on 

 surface harmonics, the equations determining the values of the 

 constant multipliers in the subsidiary terms required are of the 

 types (36^), (37,,) or (38 4 ) to (41,,) — the latter including the case 

 i = 0. The quantities on the right-hand sides of these equations, 

 answering to R i} Ti etc., are deduced in the same way as the 

 corresponding quantities for nearly spherical solids were deduced 

 in my paper in the American Journal. The only mathematical 

 difficulty consists in representing products of spherical harmonics, 

 and products of their differential coefficients &c, as simple spherical 

 harmonics or differential coefficients of simple spherical harmonics. 

 The necessary results of this kind for the shell problems analogous 

 to the single surface problems I treated in the American Journal 

 already exist there. When the quantities answering to Rt, Ti &c. 

 in (36,,) to (41^) are known, nothing remains but to substitute 

 them for R it T i} &c. in the general formula?, such as (91,,). 



