I 
THE MECHANICAL EQUIVALENT OF HEAT. 
e><k — 
if' 
A 
and 
From this pair of equations we can eliminate obtain 
2[e^(«2-o _ _ X] = 0 
bOl 
( 5 ) . 
( 6 ) .* 
( 7 ). 
which is the equation from which to determine X. 
So far as we are aware, the solution of these equations cannot be exjiressed as an 
algebraical expansion, except on the assumption that Xq, Xq, XT are small. Making 
this assumption, and writing x = — 2q, successive ap^Droximation leads to 
200 
( 8 ). 
The method which we have found best adapted to the numerical solution of such 
equations amounts practically to tracing their graph, using, for instance, in equation 
(7), X and /(X) as abscissa and ordinate, A rough approximation to the value of X is 
soon obtained, and afterwards by means of a table of logarithms as close an approxi¬ 
mation as is necessary can be found. 
Since [x does not occur as an exponential, its value is at once found from (5) or (6). 
Equation (4) can be solved for T in a similar manner to equation (7), using T and 
/(T) as abscissa and ordinate. A rough approximation to the value of T is given by 
equation (8), 
T = + i ((3 - 2(,). 
The forms given to equations (4) and (7) are those which we have found most con¬ 
venient from which to calculate the values of/"(X) and/’(T). 
Besides being thus able to eliminate radiation, we can at the same time eliminate 
the change in all those quantities whose rate of variation can be expressed as a linear 
function of the temperature. Thus, on p. 477, we have shown that when the 
variations in M and a are considered, their full values are 
. * Equation (6) shows that Oq is not altogether at our disposal, but must be less tha.n ///X. For, 
otherwise, the calorimeter would reach its final state, where the loss by radiation balances the supply of 
heat, and could never attain the temperature + 
