ON THE MECHANICAL EQUIVALENT OF HEAT. 
349 
Thus, taking the mean capacity of water between the temperatures of 32° and 
212° as the standard capacity, the mean specific heat between T^° and 212° would be 
1 -h X = 1 + 0-000028 (Ti° - 32) ; 
and, if Ti° is the mean initial temperature of the water of any number of trials, 
1 + X is the mean specific heat of the water in all the trials. The mean specific 
heat of the difference of two trials would be 1 + X ; this appears as follows :— 
Suppose 1 + X^ to be the mean specific heat for a set of heavy trials, and Wj the 
mean weight of water, and (1 + X^) to be mean specific heat of a corresponding set 
of light trials, and Wg the mean weight of water, T^®, T 2 ° being respectively the 
initial temperatures of and Wg, the difference of the total heats would be 
(1 + X,) (212 - IV) W, - (1 + X.g) (212 - Tg) Wg, 
and the mean specific heat would be approximately 
(212 - T,°) W, - (212 - T°) Wg + 180 - XAVg) _ 180 (yW^ - ^ 
(212 - V) Wi - (212 - Tg) Wg ~ 180 (W^ - Wo) ’ 
and, as in the heavy and light trials = 2Wg approximately, the mean specific 
heat by Regnault’s formula would be 
1 + 2Xi - Xg = 1 + 0-000028 [2 (Ti - 32) - (Tg - 32)]. 
This result is obtained by merely summing the trials, but counting the water in 
the light trials as negative. 
X = -000028 2 
W (V - 32) 
2 (W) 
The Gradual Rising of the Indices of the Thermometer. 
52. Where, as is generally the case, the indices of the thermometers are gradually 
rising, if they are used between the intervals at which they are corrected, the last 
observed correction being applied, there will be an error which will be negative, 
and of magnitude equal to the rate of rise during the interval multiplied by the 
interval. Thus, if the trials are uniformly distributed between the intervals of 
correction, the correction would be O'5a, where a is the observed rise in the interval, 
hence the relative correction on the equivalent, taking and a^, as the mean rises 
between the intervals of correction of the initial and final thermometers, would be 
