OjST the mechanical equivalent of heat. 311 
To a second approximation the rate of work with varying speed is 
Mw = [ 1 + 2a2 cos H (^ - - Th) + cos 2 h (^ - - To)} [ 1 + %cos?i (^ ~ Tj)} (4), 
and from this it appears that the mean rate of work is 
cooMo (1 + cos nT^, 
which shows that the relative error in taking this as MqWq is -j- cos nTc^. Thus the 
error arising from fluctuations in speed of 2(:rw is within the limits i when the 
resistance varies as the square of the speed, as in the hydraulic brakes. 
Where, as in the brake under consideration, there is an automatic adjustment, by 
wdiich the quantity of water in the brakes is adjusted to the speed, so as to maintain 
the resistance constant, there will be no error caused by such gradual variations of 
speed as result from changes in the boiler pressure, since the automatic adjustment 
can keep pace with them. But it takes time for the water to get in and out, and any 
variations, so rapid that, owing to the inertia of the brake case with its load, their 
effect has been reversed before the case has moved sufficiently to affect the water in 
the brake, will produce errors. 
Such cyclic variations of speed attend all motions derived from reciprocatiug engines, 
and it is only these, and not the secular variations, that produce errors. 
The Variations in the Speed oj Rotation oj the Steam Engine. 
8. The cyclic variations all go through one or two complete periods in the time of 
revolution of the engine, and are approximately simple harmonic functions of the 
time. 
They arise from three distinct causes :— 
(1.) The varying energy of motion of the reciprocating parts ; 
(2.) The varying moment of the effort of the steam pressures on the cranks ; 
(3.) The effect of gravitation on the unbalanced parts in the engine. 
In the case of a simple vertical engine, unbalanced and working with moderate 
expansion, these variations of speed may be severally estimated when I, the moment 
of inertia of the revolving parts, r the half-stroke of the reciprocating parts, and W 
the weight of these parts are known together with N the number of revolutions jjer 
minute, and U the work done per stroke. 
For, considering the variations as existing separately, we may assume that the 
angular motion would be steady but for the particular effect, thus : 
