198 Mr. C. V. Bovs on Apparatus 



except one of lateral translation, therefore the combination 

 c, F, D, B, and B x will move longitudinally till the central 

 distances of A Aj are proportional to their speeds ; but, unfor- 

 tunately, by this time the obliquity of the block D has become 

 a maximum, and therefore its rate of longitudinal travel is a 

 maximum also. It therefore travels on and introduces an 

 error of position on the opposite side, which is corrected as 

 before; and so the frame F oscillates on either side of its cor- 

 rect position. The motion may be considered more exactly 

 in this way: — The rate at which D twists is proportional to 

 the error of F, while the rate at which F corrects its position 

 is proportional to the amount of twist of D: this expressed 



mathematically gives the equation -=-^ = — y } the solution of 



which is y — sin x or y = cos x. The motion of both D and F 

 therefore is harmonic. An exactly analogous case is that of 

 a heavy body moving under the influence of a force which 

 varies with the displacement, the movements of F being equi- 

 valent to that of the heavy body, while the rate at which D 

 twists represents the force. Of course in the dividing-machine 

 described we have nothing to do with inertia or with a resist- 

 ing force ; but the movements are the result of an action of 

 pure rolling, and it so happens that they follow exactly the 

 same law as the movements of a vibrating heavy body. This 

 view of the subject removes all difficulty in discovering what 

 will be the action during either slow or very rapid variations 

 in the ratio of the velocities. Let the dynamical equivalent 

 be a heavy magnetic needle balanced in a magnetic field. 

 Then the direction of the field corresponds to ratio of veloci- 

 ties, and the motions of the needle to the motion of the 

 frame F. As has been shown in the case of a constant ratio, 

 an error of position of F at starting causes F to oscillate on 

 either side of its correct place, just as a displacement of a 

 magnetic needle causes it to oscillate. Now suppose F to be 

 in its correct position and then the ratio to change slowly (that 

 is, slowly compared with the time of an oscillation); then the 

 frame F will move slowly also, always being in the correct 

 position, because during a slow change in the direction of the 

 magnetic field a balanced needle follows the change without 

 oscillation. Next suppose the ratio to alternate rapidly — that 

 is, rapidly compared with the time of an oscillation^ as, for 

 instance, is the case when a steam-engine is made to do work 

 uniformly, for work is put in intermittently and taken out 

 gradually; then the dividing-machine will steadily show the 

 mean value, and will take no account of the rapid variations 



