46 
Proceedings of Royal Society of Edinburgh. [jan. 7, 
Q has attained half its final value. The point x where this con- 
dition is fulfilled at time t may he called the mid-front of the 
procession. It travels at the velocity or half the wave- 
velocity ; which agrees with the result of Stokes. 
We may arbitrarily define “the front” as the succession of 
augmenting waves which pass between the times corresponding to 
m = +10 and ??i= -10 (or any other considerable number instead 
of 10). Thus the time taken by the front, in passing the place 
x = n\, is 40w _1 The space travelled by the mid-front in 
this time is 20 yw -1 which may, arbitrarily, be defined as the 
length of the front. It increases in proportion to Jn ; and there- 
fore in proportion to Jt, as said above. The effect upon phase of 
the changing waves in the front ; due to the fluctuations of e, and 
to the law of augmentation of Q from zero to its final va lue ; is to 
be illustrated by calculations and graphic representations, which I 
hope will be given on a future occasion. 
The rear of a wholly free procession of waves may be quite readily 
studied after the constitution of the front has been fully investi- 
gate, by superimposing an annulling surface-pressure upon the 
originating pressure represented by (12) above, after the originating 
pressure has been continued so long as to produce a procession of 
any desired number of waves. 
2. Numerical and other Additions to his Paper, read on 
6th December 1886, on the Foundations of the Kinetic 
Theory of Gases. By Professor Tait. 
In the case of diffusion, in a long tube of unit section, suppose 
that we have, at section x of the tube, w^s and n 2 P 2 s per cubic 
unit, with translational speeds cq and a 2 , respectively. If Gj be 
the whole mass of the first gas on the negative side of the section, 
it is shown that the rate of flow of that gas is 
Obviously 
