Ox THE MECHANICAL EQUIVALENT OF HEAT 401 



instantaneously. To estimate t , T a being the temperature of the air, 

 we have, according to Newton's law of cooling, 



t 



C(t- Q = _ T C(0 T a } dt nearly, 



~ a / 



0" 0' 

 t = c tf , _ T nearly, 



ri 



where it is to be noted that -,, _ is nearly a constant for all values of 



" *- a 



0" T a according to Newton's law of cooling. 



The temperature reaches a maximum nearly at the time 



0"o' t 



and if 6 m is the maximum temperature, we have the value of 0" as 

 follows : 



0" = T" = 0^ + C(t m + cL): 



\. m ' v/ 7 



and this is the final temperature provided there was no loss of heat. 



When the final temperature of the water is nearly equal to that of 

 the air, C will be small, but the time i m of reaching the maximum 

 will be great. If a is a constant, we can put C = a (6" T a ), and 

 G(t n + c ) will be a minimum, when 



or a = - 



ac 



That is, the temperature of the air must be lower than the tempera- 

 ture of the water, so that T a = 6" as nearly as possible ; but the for- 

 mula shows that this method makes the corrections greater than if we 

 make T a = d', the reason being that the maximum temperature is not 

 reached until after an infinite time. It will in practice, however, be 

 found best to make the temperature of the water at the beginning 

 about that of the air. It is by far the best and easiest method to 

 make all the corrections graphically, and I have constructed the follow- 

 ing graphical method from the formula?. 



First make a series of measurements of the temperature of the water 

 of the calorimeter, before and after the basket is dipped, together with 

 the times. Then plot them on a piece of paper as in Fig. 5, making 

 the scale sufficiently large to insure accuracy. Five or ten centimeters 

 to a degree are sufficient. 



nab c d is the plot of the temperature of the water of the calori- 

 26 



