MR. J. H. JEANS ON THE DISTRIBUTION OF MOLECULAR ENERGY. 
417 
Let us call encounters of this type, encounters of class A, and denote their number 
(expression (v.) ) by N^. 
There will be a second class of encounters which will lie called class B, such that 
the co-ordinates after the encounter lie within the limits (iil.), (p. 41G). The 
co-ordinates before the encounter will accordingly lie within certain other limits, 
(Ip dV (lu dp dP' du' dx" df dz" . 
surrounding certain values p, P . . . &c. By Liouville's Theorem, tlie complete 
diflerential (vi.) is equal to the complete differential (iii.), hence the number of 
encounters of class B will, by compaiTson with (v.), be seen to be 
e-” FF 77 dp dP du dp dP' dll dx" df dz". 
the positional co-ordinates and therefore also remaining unaltered by the 
encounter. 
Let No be the total number of molecules lying within a range dP. Then a 
certain number of encounters of class A will result in a unit lo.ss to N^, a certain 
number of encounters of cla.ss B in a unit o-aln. Thus, if for one at least of the 
co-ordinates which are changed by the encounter, say we have where A 
denotes an increase due to an encounter, then it is certain that the co-ordinate ^ will 
be placed outside the limit d^, and No will accordingly be diminished by unity. 
It is, however, conceivable that for every co-oixlinate we may have Af <d^, and in 
this case there is a prol)ability \ that no single co-ordinate passes outside its limits, 
and therefore that the molecide after encounter must still Ije counted in No. It is 
easy to see that the probaldllty that the ^ co-ordinate remains with the limits cf is 
( 1 )) and therefore that 
\ = n 1 - 
where IT denotes continued multiplication extending to all the <p s co-ordinates. The 
I 0 .SS experienced by Nq on account of encounters of class A will therefore be (1 —■ X) Na 
where X = 0 for certain values of x" y" z", and is a proper fraction over the remainder, 
and the boundary of these regions depends on the co-ordinates of encounter. 
It follows tliat as the result of encounters of classes A and B com1)ined, there is a 
net gain to Nq of 
[(1 _ FP' - (1 - X)c->> FF']#' dp dP du dp dV du dx” dy” dz". 
Tlie expression in square Ijrackets may Ije written as A {(1 — X)e~’'FF'}. Hence 
the total gain to N^ arising from all classes of encounters will be 
dP 
>A{(1 - X)c 
FF'j ff dp du dp' c7P' du dx” dy" dz". 
(vii.). 
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II 
VOL. CXCVI.-X. 
