206 Prof. Challis on the Hydrodynamical Theory of 



the acceleration of the sphere by the proper condensation of the 

 undulation has a constant ratio to the acceleration of the fluid 

 where the sphere is situated. Now,, since the fluid is just as 

 much accelerated in the condensed as in the rarefied parts of the 

 wave, the positive and negative velocities being exactly equal 

 (art. 27), it follows that the sphere, not being of variable den- 

 sity like the fluid, is acted upon by greater accelerative forces 

 when in the condensed part of a wave than when it is in the 

 rarefied part. Since, however, for each part the positive accele- 

 rative forces are exactly equal to the negative, there is no residual 

 accelerative action in either direction, and therefore no tendency 

 to produce permanent translation. 



35. I have ascertained by previous researches that the con- 

 densation due to the reaction of the sphere is, to the first ap- 

 proximation, so distributed that at each instant there is as much 

 condensation on one half of the surface as rarefaction on the other 

 half, and that the}' are similarly distributed about the axis. iVlso 

 the variation of condensation at any given point of the surface 

 follows the law of the incident undulations, so that the resulting 

 accelerative action on the atom is i^eriodic. When, however, it 

 is considered that the reaction due to the incidence of the con- 

 densed portion of a wave must be greater than that due to the 

 rarefied portion, quautities of the second order being taken into 

 account, it will be seen that the accelerative action in the one 

 case is not exactly compensated for by that in the other, and that 

 there may be a residual action tending to produce permanent 

 transfer. The eff'ect will be extremely small on account of the 

 very small condensations produced by the reaction, and may be 

 considered to be taken into account by the process about to be 

 applied to the distribution of density due to the third cause, 



36. The law of distribution due to the third cause is inde- 

 pendent of the time, being determined by the relative magni- 

 tudes of b and X, the breadth of the undulations, or it may be 

 by X only. Hence, omitting periodic terms contained in f [oo, t), 

 which do not apply in the present research, and assuming, for 

 reasons already given, that this function is equal to an unknown 

 constant H, we obtain 



d^_ 

 de """ 



At another epoch, and for a different value of the condensation, 

 we might have 



dV, _ ^^V/H ^ 



for although the epoch be different, dt may be assumed to be 



