164 PROCEEDINGS OF THE AMERICAN ACADEMY 



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 d 6 the angle moved by the shaft. The work done in the time t 

 will be f M d 6. If the moment of the force is constant, the integral 

 is simply 316; but it is impossible to obtain an engine which funs 

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

 integral, 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 follow- 

 ing 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 



M=C (£)•, 

 where C is a constant. Hence the work done becomes 



of(ffi*t= cf&Jj u 



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

 torsion circle, we can assume in this investigation that the mean 

 velocity is constant, and equal to v Q . The form of the variation of 

 the velocity must be assumed, and I shall put, without further dis- 

 cussion, 



2*V 

 dt 



= v (l +CCOS -^) 



We then find, on integrating from a to 0, 



w— Cv n 3 a (1 + 1 c 2 ), 



which is the work on the calorimeter during one revolution of the 

 engine. 



The equation of the motion of the calorimeter, supposing it to be 

 nearly stationary, and neglecting the change of torsion of the sus- 

 pending wire, is 



m d 2 if, WD , n . /, - 2*<\s A 



- -r-ff — — « r Uv- ( 1 4- c cos - - ) = 0, 



g d t- 2 ' ° V. ' a ) 



where m is the moment of inertia of the calorimeter and its attach- 

 ments, \\i is the angular position of the calorimeter, W is the sum of 



