MR. J. H. JEANS ON THE DISTRIBUTION OF MOLECULAR ENERGY. 
413 
(ii.) that an encounter is similar to an impact in the former problem, in that it 
may or may not entirely change the translational velocities of the two 
molecules concerned, but that the internal velocities are only changed by a 
small amount. 
The sudden increase in any quantity f consequent on an encounter will be denoted 
by A^. 
In virtue of the above assumptions 
= 0, Ar = 0, 
Af^ and As are small, and Aw will in general be comparable with u. 
A “collision” wdll be a special case of an encounter, and may be described as 
follows. Suppose that every molecule is surrounded by a small sphere, of which the 
centre coincides with the centre of gravity of the molecule, and which moves as 
though it were rigidly attached to the molecule. The radius of the sphere is not 
yet fixed, but it must be such that the s])here entirely encloses the matter ot which 
the molecule is composed. Tlien a collision will be defined as an encounter which is 
such that the spheres of the two molecules which are engaged, intersect; the 
“ duration ” of a collision will be taken to be sufficiently long to include the whole 
interval from the instant at which the spheres first intersect to the instant at which 
they separate. The assumptions as to the nature of the gas, which are usually 
expressed by saying that the gas is molekular-iitigeordnet, and that the number of 
collisions in which three or more molecules are engaged, is infinitely small in com¬ 
parison with the number of binary collisions, will be replaced by the following 
assumptions : 
(i.) The duration of a collision is so short, that the positional co-ordinates may be 
treated as constant throughout the collision, while the velocity co-ordi¬ 
nates are abruptly changed. 
(ii.) The chance of any molecular sphere intersecting two other spheres at once, 
vanishes in comparison with the chance of its intersecting one other sphere. 
(iii.) The chance that a molecule A is found with all its co-ordinates within certain 
small ranges of values, which are such that the sphere of the molecule does 
not intersect any other sphere, depends solely upon the co-ordinates of the 
molecule A, and upon the potential upon A of the field of intermolecular 
force; it does not depend upon the arrangement of the other molecules. 
The Charaeteristic Equation. 
§ 19. Starting from the state of the gas at the time t = 0, we can arrive 
at the state after an interval dt, by imagining the following succession of 
events. Suppose in the first })lace that each molecule is allowed to move under 
