Finite Integrals of certain Partial Differential Equations. 213 



to see what mechanical objection can be urged against this 

 realization of the problem, which is extremely simple. 



12. The theory of " action at a distance " being rejected, 

 which is necessary in order to explain the facts at all, the 

 effects of gravity can in principle be referred to only two 

 conceivable causes. The tendency of two molecules of matter 

 to approach each other can be referred (1) to a motion pos- 

 sessed by the molecules themselves disturbing the equilibrium 

 of pressure of the medium between them ; (2) to a motion 

 possessed by the medium itself (in the form of streams or 

 currents) acting upon the molecules. The first of these two 

 conditions appears to be inadmissible, from the fact that we 

 cannot interfere with or modify gravity at w^ill, whereas we 

 can very readily interfere with or modify the motion of the 

 molecules of matter (as by adding or subtracting heat, for ex- 

 ample). It therefore would appear that gravity must be due 

 to some motion that we cannot interfere with, i. e. to a motion 

 in the external medium which we cannot handle or which is 

 beyond our control. Only one conclusion appears therefore 

 to be possible here ; and therefore it would seem that the theory 

 of Le Sage can scarcely be regarded as a mere hypothesis, 

 but rather as an irresistible deduction which is forced upon us 

 in the absence of any other conceivable inference. Certainly, 

 if simplicity be a recommendation, the theory needs no recom- 

 mendation on that ground. 



London, July 1877. 



XXYI. The Finite Integrals of certain Partial Differential 

 Equations which present theihselves in Physical Investiga- 

 tions. By the Rev. S. Earkshaw, M.A. 



To the Editors of the Philosophical Magazine and Journal, 

 Gentlemen, 



THE Astronomer Royal, in his treatise ' On Sound and 

 Atmospheric Vibrations,' has drawm particular attention 

 to equations of the following general form, 



d'^u __ cPu a du 



de''d^'^xd^' ^^) 



and has expressed an opinion that equations of this class can- 

 not be approached by any one general method of attack. The 

 equation has been solved by the Astronomer Royal himself in 

 a finite form for the two cases a = and a = 2 ; but in other 

 cases he has had recourse to infinite series, which, it is observed, 



