438 Mr. J. Prescott on the Precise Effect of Radial 



the additional displacements due to the harmonic forces, and 

 it might appear that these would be the displacements repre- 

 sented by u, v y w above. But i£ the radial forces are very 

 much larger than the harmonic forces, it is obvious that they 

 have the effect of reducing the deviations from the mean 

 sphere. When any spherical shell is distorted from the 

 spherical form the particles of the shell are in places where 

 the radial forces are unequal, and thus differential forces are 

 brought into play which are of a similar type to those of the 

 harmonic forces. 



The difficulty arises from the fact that a, y, and z are used 

 with double meanings. In the expressions for the potential of 

 the external forces and for those forces themselves, they mean 

 space- coordinates. Thus if V denotes the potential of the 

 external forces, we suppose V expressible in terms of these 

 space-coordinates thus 



V=F<>, *,,*), 



and the forces are derived from this by differentiation and 

 expressible in the form 



Now there is a second meaning to a, y, and z. They are used 

 as the distinguishing coordinates of a particle of the elastic 

 body even after the body is strained. They are the space- 

 coordinates of the position of the particle before strain. The 

 space-coordinates after strain are represented by x + n, y + v, 

 and z + w. It is clear, therefore, that the x- force at the point 

 occupied by the particle whose distinguishing coordinates are 

 (%, y, z) is, after the strain, ¥ x (x + u, y + v, z + w). Now 

 these external forces at the point (x + u, y + v, z + w) are in 

 equilibrium with the stresses at that point. But the stresses 

 are regarded as functions of the distinguishing coordinates 

 of the particles of the body; so that if P = <p(x, y, z) is one 

 of these stresses, we mean by <£ (jb 9 y, z) the stress at that 

 point of the body which was at (x, y, z) before strain, and 

 which is at (x + u, y + v, z + w) after strain. Thus in the 

 equations of equilibrium we must express the stresses in 

 terms of (a, y, z) and the external forces in terms of (x + u, 

 y + v, z + w), and that is my method. 



In most problems the external forces in the displaced 

 positions would differ so little from those in the original 

 positions, that it would be superfluous to take account of the 

 difference. But in the problem considered here the differ- 

 ence in the case of the radial forces is of the same order of 



