1P4 Lord Kelvin on the Motion of Ponderable Matter 



§ 12. The first dynamic question that occurs to us, return- 

 ing to the supposition of moving atom and of ether outside it 

 at rest, is : — What is the total kinetic energy («) of the 

 portion of the ether which at any instant is within the atom ? 

 To answer it, think of an infinite circular cylinder of the 

 ether in the space traversed by the atom. The time-integral 

 from any era t = of the total kinetic energy of the ether in 

 this cylinder is t/c ; because the ether outside the cylinder is 

 undisturbed by the motion of the atom according to our 

 present assumptions. Consider any circular disk of this 

 cylinder of infinitely small thickness e. After the atom has 

 passed it, it has contributed to t/c, an amount equal to 

 the time-integral of the kinetic energies of all the orbits of 

 small parts into which we may suppose it divided, and it con- 

 tributes no more in subsequent time. Imagine the disk 

 divided into concentric rings of rectangular cross-section e dr\ 

 The mass of one of these rings is ^irr'dr'e because its density 

 is unity ; and all its parts move in equal and similar orbits. 

 Thus we find that the total contribution of the disk amounts to 



2ire V dr'r' W/'dt (12), 



where \ds 2 ldt denotes integration over one-half the orbit of 

 a particle of ether whose undisturbed distance from the 

 central line is ?■' ; (because ^ds 2 /dt 2 is the kinetic energy of an 

 ideal particle of unit mass moving in the orbit considered). 

 Now the time-integral Kt is wholly made up by contributions 

 of successive disks of the cylinder. Hence (12) shows the 

 contribution per time e/g, q being the velocity of the atom ; 

 and (k being the contribution per unit of time) we therefore 

 have 



K = 2irq\ 1 drr'\dsydt . . . . (13). 



§ 13. The double integral shown in (13) has been evaluated 

 with amply sufficient accuracy for our present purpose by 

 seemingly rough summations ; firstly, the summations §ds 2 /dt 

 for the ten orbits shown in fig. 4, and secondly, summation of 

 these sums each multiplied by dr' V. In the summations for 

 each half-orbit, ds has been taken as the lengths of the curve 

 between the consecutive points from which the curve has 

 been traced. TIhs implies taking dt=l throughout the three 

 orbits corresponding to undisturbed distances from the central 

 line equal respectively to 0, "6, "8 ; and throughout the other 

 semi-orbits, except for the portions next the corner, which 

 correspond essentially to intervals each < 1. The plan 



