40 
MR, J. J. WATERSTON ON THE PHYSICS OF MEDIA COMPOSED OF 
velocities ; we have therefore to enquire into the effects of this diversity upon the 
velocity of transmission. 
The molecules in a small sphere of the medium at any given instant may he classed 
in respect to their velocity into sets, and of each set we are allowed, by the hypothesis 
which we are following, to assume that there is an equal number moving in every 
direction, and that since continuous uniformity in the density requires that the 
number contained in the spherical space should be always the same, the exit of one 
of a set may be conceived to be immediately followed by the entrance of another of 
the same moving in the same direction and with the same velocity. One-half the 
number in a set is increasing their distance from a given plane, and the other half 
diminishing their distance. Let the motions of one of these halves be resolved in the 
direction perpendicular to this plane, and let us add together such resolved spaces as 
are described by all the molecules of the set that happen to be in the sphere during a 
constant time for so long as they remain in it, and divide by the constant number of 
molecules of the set in the sphere at all times; the quotient must evidently be the 
mean velocity in that direction and set, and must be the uniform rate with which an 
impulse is conveyed in one direction by means of an infinite series of impacts, the 
space between two impacts measured in the constant direction being the step forward 
made by the infinitesimal portion of the impulse contained in the traversing motion of 
the molecule from one of the impacts to the other. 
The number moving in any one direction with the velocity u is equal to the 
number moving in any other direction with the same velocity, and each of these 
numbers takes the same time to traverse the sphere. If we compare this time with 
that taken by the molecules of another velocity or set, it is obvious that these times 
must be inversely as the velocities, and the number that continuously pass th rough the 
sphere or any other constant space in a constant time must be as the velocity : for this 
may be estimated as if there were continuous currents of molecules moving in every 
possible direction with the respective velocities ; the encounters that may be imagined 
to interrupt this continuity being infinite in number do not alter the general average 
of velocity or direction or proportionate number, and therefore each velocity and 
direction may be taken as constant. Now, for any one velocity u viewed thus as 
constant, the mean resolved velocit}^ in one direction of all the molecules that happen 
to move with this velocity at any instant is, as above demonstrated, equal to \u; 
but if ive acid up the resolved spaces traversed by cdl the molecides that have been in 
the sphere with this velocity during a constant time, and divide by the constant number 
that are in the sphere at any instant, we require to multiply \u by a factor that is 
proportional to u, so that \id, the resulting product, is proportional to the mean 
distance traversed in a constant time by the molecules that have appeared in a 
constant space to move with this velocity during the constant time. 
Now, suppose an impulse to be given to the medium at any point, and an 
indefinitely long cylinder of the medium to extend from this point ; the impulse 
