408 Lord Kelvin on the Front and Rear of a 
surface at time 257 after a commencement correspondingly 
compounded from fig. 8, and another similar figure drawn to 
represent the rear of the finite (two-ended) procession which 
we are now considering. 
§ 30. Direct attack on the problem thus indirectly solved, 
gives, for the case of 1000 wave-crests in the beginning, the 
following explicit solution, 
=. 
—t= = (-D¥(e-42) . . 
i=0 
where wv is a function found according to the principles 
indicated in § 4 above, to express the same surface-displace- 
ment as our function ¢ of § 12, and the proper velocities 
below the surface to give, in the sum, a rightward procession 
of waves. Our present solution shows how rapidly the initial 
sinusoidality of the head and front of a one-ended infinite 
procession, travelling rightwards, is disturbed in virtue of 
the hydrokinetic circumstances of a procession invading still 
water. Our solution, and the item towards it represented 
in figs. 6 and 7, and in fig. 2 of § 6 above, show how 
rapidly fresh crests are formed. The whole investigation 
shows how very far from finding any definite “ group- 
velocity” we are, in any initially given group of two, three, 
four, or any number, however great, of waves. I hope in 
some future communication to the Royal Society of Edinburgh 
to return to this subject in connexion with the energy principle 
set forth by Osborne Reynolds *, and the interferential theory 
of Stokesf and Rayleigh{ giving an absolutely definite 
group-velocity in their case of an infinite number of mutually 
supporting groups. But my first hydrokinetic duty, the per- 
formance of which I hope may not be long deferred, is to 
tulfil my promises regarding ship-waves, and circular waves 
travelling in all directions from a place of disturbance in 
water. 
§ 31. The following tables show some of the most important 
numbers which have been calculated, and which may be useful 
in farther prosecution of the subject of the present paper. 
* Nature, vol. xvi. 1877, pp. 843-344, 
+ Smith’s Prize Paper, Camb. Univ. Calendar, 1876, 
t ‘Sound,’ ed. 1, vol. i. 1877, pp. 246-247. 
ee Bias - 
