Attractive and Repulsive Forces. 205 



half of the surface would be much diminished by reason of their 

 limited lateral divergence on being transmitted beyond the first 

 half. In extreme cases a portion of the fluid in contact with the 

 second hemispherical surface might be altogether undisturbed. 

 On the other hand, if waves of large breadth were incident on a 

 small sphere, the condensations might become by lateral diver- 

 gence greater on the second half surface than on the first, because 

 the conditions of the motion would approximate to those of 

 spontaneous motions along an axis, in which the condensation 

 corresponding to a given velocity is greater than in plane- com- 

 posite waves in the ratio of a^'^ to a^'. Accordingly, so far as the 

 action of the spontaneous vibrations takes effect, the condensa- 

 tion on the surface might either decrease or increase as x is 

 greater; but in the actual physical circumstances of the sether 

 and the atoms the gradation must always be extremely small j 

 otherwise, by reason of the vast elastic force of the aether, the 

 acceleration of the atom would exceed the amount which expe- 

 riment and observation appear to indicate. 



33. This being understood, let 6 be the angle which any 

 radius makes with the part of the axis on the negative side of 

 the sphere^s centre, so that, if x^ be the abscissa of the centre, 

 x=Xj — bcos6y b being the radius of the sphere. Also let 

 p — a^'^p = a^\ 1 + cr). Then we have with sufficient approximation 



(T—dJiXy t) = crj[x—h cos 6, t) = o-J{x^, t) — a^f [x^, t)b cos 6, 



because, from what is said above, the value off\x, t) must be 

 very small and very nearly constant. Consequently the whole 

 pressure on the sphere estimated in the positive direction is 



^^ira'^ab sin 6 cos 6^ ^6>, from ^ = to ^ = tt. 



On substituting the above value of cr this integral will be found 

 to be 



and consequently, if A be the ratio of the density of the sphere 

 to that of the cether, 



the accelerative force = ir^f'i^} 0^ 



which is positive iif'{x, t) is negative — that is, if the condensa- 

 tion decreases as x is greater. 



34. With respect to the composition of the function /, so far 

 as it depends on the second of the causes of distribution of con- 

 densation on the surface of the sphere mentioned in art. 325 it 

 will contain only periodic terms, because the action of this cause 

 is periodic. In fact it may be shown, just as in art. 19, that 



