438 Mr. J. H. Jeans on the 



ideal atom in such a way that, if we imagine the two atoms 

 " superposed " (made to occupy the same space), then the ion 

 A will be near to the element A', B to B', and so on, while at 

 the same time the ideal atom shall satisfy the condition of 

 perfect spherical symmetry. 



In order to avoid a discussion of continuity at points at 

 which the electrical density changes sign, we may suppose 

 elements for which p is of different sign to be separated by a 

 thin transition-layer of thickness small in comparison with the 

 dimensions of an element. 



Imagine the ideal atom subjected successively to a system 

 of displacements finite in number, say 



u=i*S t , A = 0, 



or u = 0, A = r*Sj, 



where A, as in § 20, denotes the tangential dilatation, S^ denotes 

 a surface harmonic of order /, and s, t have any number of 

 integral values. For each of these displacements calculate 

 the forces acting upon each of the elements A 7 , B', . . . . ; these 

 will be linear functions of C p C 2 , . . . . , the constants which 

 may be supposed to enter in the specification of the law of 

 force in the ideal atom. In order that the forces calculated 

 in this way may be finite, it is necessary and sufficient that 

 the forces between elements at a small distance r shall be of 



an or 



der not greater than ^ in ( -)■ 



Imagine the real atom to undergo the same system of dis- 

 placements, and for each of these displacements calculate the 

 forces acting upon each of the ions in terms of the constants 

 which occur in the actual law of force between ions (sup- 

 posed known) . Now equate the forces acting upon A to the 

 forces acting upon A / in the case of the same displacement. 

 We see in this way that the forces acting on A, B, C . . . . can 

 be made identical with the forces acting on A', B', C . . . . for 

 the case of any finite number of displacements (the number 

 being as large as we please so long as it is not infinite) pro- 

 vided that the infinite number of constants C 1; C 2 . . . . satisfy 

 afinite number of linear algebraical equations. 



Corresponding to any selected system of displacements it 

 will be possible to find C 1? C 2 . . . . , so that the forces acting 

 upon corresponding elements and ions may be equal in every 

 case. Cases of failure (cases in which two or more of the 

 equations for C are inconsistent with finite values for C) 

 can always be avoided by slightly altering one or more of the 

 selected displacements. If the displacements are suitably 

 chosen, it will be possible very approximately to expand any 



