338 
MB. J. J. THOMSON ON SOME APPLICATIONS OF 
but by equation (97) we see that 
dm n dn 
a =,w ; / 3 =^ 
dd 
so 
that 
M=i ^(m+n)Se s . . 
(100) 
( 101 ) 
so that if the coefficient of elasticity diminish with the temperature, as is the case for 
most substances, an increase in the strain will produce a lowering of temperature 
which can be calculated by equation (101); if, on the contrary, the coefficient of 
elasticity increase with the temperature so that d(m-\-n)/d0 is positive, an increase in 
the strain will cause the temperature to rise. These results were given by Sir 
William Thomson in his paper on the “Dynamical Theory of Heat” (‘Mathematical 
and Physical Papers/ xlviii., § 202). 
§ 10. The next term in order is the coefficient of the squares of the differential co¬ 
efficients of the coordinates which fix the strain configuration. From the way this co¬ 
efficient arises we can see that, in the present state of our knowledge, there is no reason 
to believe that these involve any of the coordinates which we are considering. For if 
rj, £, are the coordinates of the centre of an element dxdydz of an elastic solid, and 
if p be the density of the element, which is supposed to be so small that its density 
may be taken as uniform, then the kinetic energy of the strained solid equals 
^p(&+vf+t?)dxdydz .( 102 ) 
Now pdxdydz is not altered by moving the body about, so that when the integration 
is completed the coefficient of w* will not involve the geometrical coordinates x. It is 
conceivable that it might depend upon the electrical coordinates, just as the coefficient 
of x 2 may depend upon these quantities ; but if this were so, the capacity of a 
condenser would be altered by making the air between the plates vibrate, and the 
time of vibration of a bar made of some dielectric would be altered by communicating 
a charge of electricity to it. This latter phenomenon actually takes place, but it is 
probably due to an alteration in the elasticity of the bar produced by the electricity, 
and not to an effect of the kind we are considering. As there is no experimental 
evidence that the coefficient of id 1 does involve the electrical coordinates, we shall not 
consider what the effects of its doing so would be. The same thing applies to the 
magnetic coordinate y. The coefficient of w 2 is evidently not a function of the 
temperature or strain coordinates. 
§ 11. The most important term containing the product of differential coefficients of 
coordinates of different kinds is the term containing the product of the rate of 
changes of the electric and magnetic coordinates. In considering this term we shall 
use the same notation as before, but it will be convenient to resolve y into three 
