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XLT. On the Precise Effect of Radial Forces in opposing the 

 Distortion of an Mastic Sphere. By J. Prescott, M.A., 

 Lecturer in Mathematics, Manchester School of Technology * . 



rfYHlS question is part of a more general one which is 

 L worked out very fully in Love's ' Theory of Elasticity * 

 (arts. 170-178). Since, however, there is an important error 

 in Love's solution, an error of principle and not of calcu- 

 lation merely, I have presumed in this paper to point out the 

 error and give the correct solution. It is all the more 

 necessary to point out the error because, not being a very 

 obvious one, it might pass a long time unnoticed in such an 

 excellent book as Love's. 



It is supposed that straining forces act on a solid homo- 

 geneous sphere which have a potential of the form V + U, 

 where Y=f(r) and U is the sum of several spherical solid 

 harmonics. Then it is reasonable to assume, as in Love, that 

 the radial displacement will be composed of solid harmonic 

 terms also. 



Let us suppose that the matter which, in the unstrained 

 sphere, was distributed over the sphere of radius r , is now 

 distributed over the surface whose equation is 



r=r 1 +.o; 



i\ being constant for each shell, and a being a small quantity 

 which is a function of i\ and angular co-ordinates. The 

 equation (1) is thus the equation of a family of nearly 

 spherical surfaces with i\ as parameter. 



Let us also suppose that the coordinates of a particle on the 

 surface whose parameter is r x are x l -\-u, yi + v, Zi + w where 

 #i? #1? z \ ai *e the coordinates of a point on the sphere of 

 radius r x . 



It will be seen from the above that we are considering the 

 displacement of every particle to be composed of two distinct 

 parts, namely (1) one in which every shell which was sphe- 

 rical in the unstrained state and had a radius /' , is strained 

 into a shell of radius r 1 ' ; and (2) displacements measured from 

 the strained spherical surface. Now the radial forces with 

 potential V will produce a radial strain, and the other type 

 of strain can be produced by the harmonic forces. 



We will now suppose that the radial forces, if acting alone, 

 would produce the radial displacement from r to r i} which, 

 we shall assume, is not sufficiently large to alter the density 

 appreciably, or to affect the elastic properties of the sphere. 

 If now we completely ignored the radial forces we could find 

 * Communicated bv the Author. 



