444 Prof. Challis on the Force of Gravity. 



In order to bring the inquiry into the laws of forces within the 

 domain of mathematics, I have added to the Newtonian hypo- 

 theses two others ; viz. that the ultimate atoms of bodies are 

 spherical, and that they are acted upon by the prcssui-e of a 

 highly elastic medium pervading space. The existence of such 

 a medium is almost demonstrated by the undulatory theory of 

 light. It is supposed that this medium is always and everywhere 

 equally in action, but that its action becomes perceptible only 

 when from any cause it is made to press unequally on opposite 

 hemispherical surfaces of the atoms. This hypothesis is in accord- 

 ance with what Faraday denominates the principle of the conser- 

 vation of force. If the physical forces are all resolvable into 

 pressures of the same medium, a reason is at once given for that 

 mutual relation and mechanical equivalence between them, which 

 experimentalists have recently begun to recognize. 



The hypotheses above stated necessitate the application of 

 partial diflFerential equations, because they require the investiga- 

 tion of the laws of the motion inter se of the different elementary 

 portions of a fluid medium. The class of dynamical questions 

 answered by common differential equations have exclusive refer- 

 ence to the motion, under the action of given forces, of a single 

 point, or of an aggregate of points rigidly connected. The great 

 problems of physical astronomy are solved by means of equa- 

 tions of this order, the treatment and applications of which are 

 well understood. It can scarcely be doubted that problems of a 

 still more comprehensive character remain to be solved by partial 

 differential equations ; but it must be confessed that at present 

 the principles of the application of the higher orders of differential 

 equations to physical questions, and the rules for drawing infer- 

 ences from their solutions, are very imperfectly known. In the 

 treatment of the problems proposed in my communication to the 

 Philosophical Magazine for November, I believe that I have 

 avoided errors of principle into which other mathematicians have 

 fallen, and that, although more expeditious and systematic me- 

 thods of solving the same problems may be discovered, different 

 results will not be obtained. There are, however, some points 

 admitting of further elucidation, which will be adverted to before 

 any conclusions are drawn relative to the force of gravity. 



The type of the waves whose dynamical action is considered 

 in Problem IV., is expressed by the equations 



\=.Kaa-=m sin( ^ [Kut — x) +c\, 



in which the origin of x may, if we please, be the centre of the 

 fixed spherical atom. In that case the values of x for points on 

 the surface of the atom are so small compared to \, that without 



