Ox THE MECHANICAL EQUIVALENT OF HEAT 431 



some time the apparatus reaches a stationary state, in which, but for 

 the radiation, the rise of temperature at all points would be the same. 

 This steady state will be theoretically reached only after an infinite 

 time; but as most of the metal is copper, and quite thin, and as the 

 whole capacity of the metal work is only about four per cent of the 

 total capacity, I have thought that one or two minutes was enough to 

 allow, though, if others do not think this time sufficient, they can 

 readily reject the first few observations of each series. When there 

 is radiation, the stationary state will never be reached theoretically, 

 though practically there is little difference from the case where there is 

 no radiation. 



The measurement of the work done can be computed as follows. 

 Let M be the moment of the force tending to turn the calorimeter, and 

 dd the angle moved by the shaft. The work done in the time t will 

 be fMdft. If the moment of the force is constant, the integral is 

 simply Mti; but it is impossible to obtain an engine which runs with 

 perfect steadiness, and although we may be able to calculate the inte- 

 gral, as far as long periods are concerned, by observation of the torsion 

 circle, yet we are not thus able to allow for the irregularity during one 

 revolution of the engine. Hence I have devised the following theory. 

 I have found, by experiments with the instrument, that the moment of 

 the force is very nearly, for high velocities at least, proportional to the 

 square of the velocity. For rapid changes of the velocity, this is not 

 exactly true, but as the paddles are very numerous in the calorimeter, 

 it is probably very nearly true. We have then 



where C is a constant. Hence the work done becomes 



n r (dov, a n r/dff\',. 



W= C I -jj- \dO = C I ( rr \flt- 



J \dt ) J \tltj 



As we allow for irregularities of long period by readings of the tor- 

 sion circle, we can assume in this investigation that the mean velocity 

 is constant, and equal to t? . The form of the variation of the velocity 

 must be assumed, and I shall put, without further discussion, 



dt 



We then find, on integrating from a to 0, 



