1899-1900.] Lord Kelvin on the Motion in an Elastic Solid. 231 
§ 1 2. The first dynamic question that occurs to us, returning 
to the supposition of moving atom and of ether outside it at rest, 
is : — What is the total kinetic energy (k) 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 — 0 of 
the total kinetic energy of the ether in this cylinder is tK ; 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 Ik, 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 contributes no 
more in subsequent time. Imagine the disk divided into con- 
centric rings of rectangular cross-section e dr . The mass of one 
of these rings is 27rr 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 
is r (because \ ds 2 Jdt 2 is the kinetic energy of an ideal particle of 
unit mass moving in the orbit considered). Hence the time- 
integral Kt is wholly made up by contributions of successive disks 
of the cylinder. Hence (12) shows the contribution per time e/q, 
q being the velocity of the atom ; and (k being the contribution 
per unit of time) we therefore have 
§ 13. The double integral shown in (13) has been evaluated 
with amply sufficient accuracy for our present purpose by 
the ten orbits shown in fig. 4, and secondly, summation of these 
orbit, ds has been taken as the lengths of the curve between the 
where fds^jdt denotes integration over one-half the orbit of a 
particle of ether whose undisturbed distance from the central line 
. . (13). 
seemingly rough summations; firstly, the summations fds* jdt for 
sums each multiplied by dr r. In the summations for each half- 
