444 Note on the preceding Paper. 



coordinates of a point of the body when in the state in which it 

 is held by the forces, not those of the same particle of the body 

 when in the state that it would have if the forces were not 

 acting. This meaning' is shown at once to be correct by 

 examining the proof of the equations of equilibrium as given, 

 for example, in the second edition of my book on ' Elasticity,' 

 Articles 44, 54. 



The incriminated solution was given in the first edition of 

 my book and omitted from the second edition, because 

 in the meantime I had found that it was unsatisfactory. 

 The problem is concerned with an elastic solid body held 

 in a nearly spherical shape by its own gravitation and 

 by external forces. The nearly spherical body is taken to 

 represent the Earth. The type of the external forces 

 is such as to include tide-generating force as a particular 

 example. In the solution in question it is assumed that the 

 stress by which the self-gravitation of the body, supposed 

 truly spherical, is balanced throughout the body, is cor- 

 related, according to Hooke's Law, with a state of strain, and 

 that this strain can be expressed by means of a displacement 

 according to the ordinary method of the theory of Elasticity. 

 According to this method the body is regarded as capable of 

 existing in two states : the first, a sphere free from gravitation, 

 and therefore also free from stress ; the second, a gravitating 

 sphere. The calculated displacement is that by which the 

 particles would pass from their positions in the first state 

 to their positions in the second state. It is essential to 

 the success of the method that the strain and and dis- 

 placement so calculated should be small quantities. When 

 the calculation is effected it is found that, unless the 

 material can be treated as incompressible, this condition is 

 not satisfied. In Mr. Prescott's notation and words, the dis- 

 placement required to change r into )\ does alter the density 

 appreciably and affects the elastic properties of the sphere. 

 The assumption in regard to the nature of the stress, by 

 which the self -gravitation of the sphere is balanced, is there- 

 fore in general untenable, and the solution fails. Mr. Prescott's 

 would fail for the same reason even if his argument which 

 is criticized above were correct. As I have pointed out in 

 the second edition of my book and elsewhere, the Earth must 

 be regarded as a body in a state of " initial stress/' This 

 view has been advanced also by Lord Rayleigh (Proc. Roy. 

 Soc. vol. lxxvii. p. 486, 1906). The solution given in the 

 first edition of my book needs correction, but not in the sense 

 indicated by Mr. Prescott. 



