208 Prof. Challis on the Hydrodynamical Theory of 



The value of cr^ at any time ^ at a given position being ex- 

 pressed by the function jx^mi-^ {- c\ we shall have 



v=^(--{"'«<)). 



showing that together with a periodic part the force ~\~ ^^ 



has a constant part, which is proper for giving to the sphere a 

 constantly accelerated motion. 



37. If the sphere, instead of being fixcd^ were free to move in 

 obedience to the accelerative force, the acceleration of its motion 

 would give rise to resistance from the inertia of the medium ; 

 but it might be showm, just as in the case of acceleration of a 

 sphere by fluid in steady motion, that the only effect would be 

 to diminish the acceleration in a certain constant ratio. 



38. If the series of incident waves be propagated from a 

 centre, /x, as is known, varies inversely as the distance from the 

 centre, and therefore ^^ and the accelerative force would vary 

 inversely as the square of the distance ^ m agreement with the law 

 of gravity. Since it is known by experience that gravity accele- 

 rates all atoms equally, in applying the foregoing formula to 

 account for the laws of gravity, it must be supposed either that 

 all atoms are of the same size, or that, for vibrations proper for 

 producing the observed effects of gravitation, the factor H is in- 

 dependent of the magnitudes of the atoms. The latter suppo- 

 sition is, I think, more likely to be true than the other, 



39. If the value of a^ be expressed by the sum of any number 



of terms such as jx^ml'^^ \-c\ we should have 



r^2 = ^ ^ 4- periodic terms;' 



from which it follows that the whole acceleration is the sum of 

 the accelerations due to the separate terms. In the proofs of 

 Propositions XV. to XVII. contained in pp. 225-239 of the 

 'Principles of Mathematics,'' I have demonstrated that those 

 condensations in different sets of vibrations which correspond to 

 non-periodic terms of the second order md^y coexist -, so that the 

 results obtained by extending the investigation to cases of the 

 propagation of different sets in different directions in space ac- 

 count generally for the coexistence of all attractive and repulsive 

 forces which are referable to vibrations of the aether. 



40. The phenomena of light indicate the existence of an un- 

 limited number of coexistent getherial vibrations for which A, has 

 every gradation of value within certain limits, and make it pro- 



