Attractive and Repulsive Forces. 199 



wliich^ as deduced mathematically, have thus far been discussed ; 

 and I hope that this supplementary treatment of the a priori 

 arguments may serve to make the explanations more intelligible 

 aud more worthy of acceptance. I proceed to the dynamical 

 action of the unsteady motions of the ?ether. 



23. This part of the inquiry is distinguished from that which 

 precedes by the circumstance that at each point the velocity and 

 its direction both vary with the time, or, if not both, that the ve- 

 locity varies. (In the ^Principles of Mathematics,^ p. 218, an 

 instance of composite vibratory motion occurs in which the di- 

 rection of the motion is constant.) Hence in the subsequent 

 application of the general equation (a) it is assumed that the 

 motion is not steady, and that consequently \udx] is always and 

 everywhere an exact differential (see art. 7). Accordingly, as i/r 

 may vary with t, we may now suppose the arbitrary function 

 %(/) in that equation to be included in a/t ; and then putting re- 

 spectively \ -~y \ ■—, ^~^ foJ^ ^j ^; ^} we have 



?«(S?-f +©=»■ •■■(•) 



An important inference is next to be drawn from this equation 

 antecedently to making use of the other general equations. I 

 have already given the reasoning proper for this purpose under 

 Proposition VII. in the Philosophical Magazine for March 1851, 

 and in pp. 186-188 of the before -cited volume. It is repeated 

 here with certain modifications, which, I think, will have the 

 effect of exhibiting more completely the logic of the argument. 



24. Since X(^'v/r)= [uda:], and by hypothesis the right-hand 

 side of this equality is an exact differential, it follows that X is a 

 function of A|r and t. Conceive a line to be drawn at a given 

 instant in the directions of the motions of the particles through 

 which it passes, and let ^, y, z be the coordinates of a point the 

 distance of which measured at the given instant along that line 

 from a given point of the same is s. Then, V being the velocity 

 at the point xyz, 



(dyfr) _ d^fr dx d-^ dy d-yjr dz 

 ds dec ds ' dy ds dz ds 



dy\r u d-^jr V dyjr to 

 "^l^'Y'^Hy'T'^lb'Y 



