474 
PROFESSOR J. J. THOMSON ON SOME APPLICATIONS OF 
and it will be convenient to begin by considering the investigations which have been 
made on this connection. 
The proof usually given is as follows (see Louth’s ‘Advanced Ligid Dynamics/ 
p. 254) 
Let us suppose that no external work is done by the system ; then— 
S(T + V) = SQ, . ..(3) 
where SQ is a small quantity of work supplied to the system. The quantity 
8 (T -f Y) occurs on the right-hand side of equation (l) : let us now consider the term 
[2 Sq clT/dq] 0 , which also occurs on the same side of the equation. If the motion be 
oscillatory, and i a period of complete recurrence ; 8 q (IT/dq will have the same value 
at the lower as it has at the upper limit of the integral, and therefore the difference of 
the values will vanish. The case when the motion is oscillatory is not, however, the 
only, nor indeed the most important, case in which this term may be neglected. Let us 
suppose that the system consists of a great number of secondary systems, or, as they 
are generally called, molecules, and that the motion of these molecules is in every 
variety of phase; then the term [2 Sq dT/dq], the sum being taken for all the 
molecules, will be small, and will not increase indefinitely with the tune, but will 
continually fluctuate within narrow limits. This is evidently true if we confine our 
attention to those coordinates which fix the configuration of the molecule relatively 
to its centre of gravity ; and, if we remember that the motion of the centre of gravity 
of the molecules is by collision with other molecules and with the sides of the 
vessel which contain them continually being reversed, we can see that the above 
statement remains true even when coordinates fixing the position of the centres of 
gravity of the molecules are included. Thus, if the time over which we integrate 
is long enough, we may neglect the term [2 Sq dT/dqJ 0 in comparison with the other 
terms which occur in equation (1), as these terms increase indefinitely with the time, 
so that in this case, even though the motion is not entirely periodic, equation (1) may 
be written— 
28 ( T dt = i 8Q, 
J o 
or 
28 = 78Q,.(4) 
where T m is the mean kinetic energy. This equation may be written— 
28 log (tT») = 8Q/T*,.(5) 
so that SQ/T,„ is a perfect differential. 
One of the ways of stating the Second Law of Thermodynamics is that SQ 6 is 
a perfect differential, 6 being the absolute temperature ; thus, if 6 is a constant 
