Gl i TRANSACTIONS OF SECTION A. 



Avhere 



A' = ^f\'^T\\''^^T-[^T\ 11'=- L'L^' 'K 



8n\Ldi/A Ids J Ld.v J J 4;r d.v dt/ 



Stt \ Idz J Lrf.i- J Ldi/ J J Air dy dz 



c'=-Lrr'^T+r'^ZT-r^n''\ r'=-jL'^v ^v 



Stt V Vd.c J Ldy A LdzA J ^ 4-n- dz d.i 

 Maxwell now solves these by taking A, B, C, F, G, H = A', B', C, F', G', 11', 

 corresponding, as he shows, and as is easily seen, to a compression along the 



lines of gravitating force, the astronomical unit of ma?s being employed, accom- 

 panied by an equal tension perpendicular to these lines, R being the resultant 

 lorce of gravitation on unit mass. This, then, he considers, is the given state of 

 stress in the ether, for which we have to account, if possible, by a corresponding 

 state of strain. This now it is impossible to effect for an arbitrary distribution 

 of gravitating matter for an ether homogeneous and either isotropic or eolotropic, 

 according to a given Greenian function. 



The difficulty will, however, be removed if we consider that we are not 

 bound to Maxwell's special solution of the equations (1) or their equivalents (2), 

 but are free to take such a solution as may be deduced from a state of strain 

 according to the laws for, say, an homogeneous isotropic ether. Such solution 

 will then yield on an element volume the same resultant force as the Maxwellian 

 stress. Now it is seen that we may make use, with slight modification, for this 

 purpose of the formula given by Lord Kelvin for an indefinite homogeneous 

 isotropic medium on which given bodily forces act per unit of mass. We thus 

 find the following displacement expressions : 



« = » 



' dV 

 p' -j^ dx'dy'dz' 



+ 



iy'-y)^ + ("-^)^ -d-vdy'dz'^ 



with analogues for v, it; where 



,. = ^{x-x'y + {y-y'y + (^^'Y, 



and the integrals are taken, throughout the space occupied by gravitating matter, 



«, n, being certain functions of X, /n for the ether. The state of stress derived 

 from these according to the laws of a homogeneous isotropic medium will be the 

 required ether stress of gravitation. 



It is evident that the ether stress corresponding to a given electrostatic field 

 can be investigated on the same lines. 



4. Conservative Systems of Prescribed Trajectories. 

 By Professor E. Odell Lovett. 



Tisserand in the first volume of his classic treatise on celestial mechanics 

 studies the problem of determining the forces under which a particle, whatever 

 be its initial position and velocity, always describes a conic section whose 

 equation is taken in its most general form. Bertrand ' was the first to set the 

 problem ; he solved it, and later Darboux and Halphen in more complete form.'^ 



' Comptes Rendns, vol. Ixxxiv. pp. 671, 731. ^ Ibid., pp. 760, 939. 



