ON CONFINED MOTION. 37 



consequently all the revolutions will be nearly isochronous, while the 

 threads or wires deviate hut little from a vertical situation.* In fact, we 

 may imagine such a revolution to be composed of two vibrations of a 

 simple pendulum, existing at the same time, in directions at right angles 

 to each other ; for while a pendulum is vibrating from north to south, it 

 is liable to the impression of any force, capable of causing a vibration from 

 east to west ; and the joint result of both vibrations will be a uniform 

 revolution in a circle, if the vibrations are equal and properly combined ; 

 but if they are unequal, the joint vibration will be ultimately an ellipsis, 

 the joint force being directed to its centre, and always proportional to 

 the distance from that centre. (Plate II. Fig. 27.) 



The near approach of these revolutions to isochronism has sometimes 

 been applied to the measurement of time, but more frequently, and more 

 successfully to the regulation of the motions of machines. Thus in Mr. 

 Watt's steam engines, two balls are fixed at the ends of rods in continual 

 revolution, and as soon as the motion becomes a little too rapid, the balls 

 rise considerably, and turn a cock which diminishes the quantity of steam 

 admitted. (Plate II. Fig. 28.) 



The same laws are applicable to many other cases of rotatory motion ; 

 for instance, if we wish to determine the height, at which a ball, revolving 

 with a given velocity, will be retained in a spherical bowl, or the incli- 

 nation of a circular road, capable of counteracting the centrifugal force of 

 a horse, running round [in] it; (for the horse, like the ball of the 

 revolving pendulum, has a centrifugal tendency, which is greater as his 

 velocity is greater;) this centrifugal force, combined with the force of 

 gravity, composes a result, which, in the case of the pendulum, is com- 

 pletely counteracted by the force of the thread or wire, and must there- 

 fore be in the direction of the thread, and which obliges the horse to 

 place his legs in a similar direction, proceeding from an imaginary 

 point of suspension above ; since he would otherwise be liable to fall out- 

 wards, if his velocity were sufficiently great. But in order to withstand 

 the pressure of the horse's legs, the road must be in a direction perpendi- 

 cular to them ; otherwise its materials will naturally be forced outwards, 

 until they produce an elevation sufficient to give the road the required 

 form. Thus, if the diameter of the ring were 40 feet, and the horse 

 moved at the rate of 12 miles an hour, he would perform about 500 revo- 

 lutions in an hour, and half a revolution in three seconds and a half. 

 Now the length of a pendulum vibrating in &J seconds, must be 39 inches 

 multiplied by the square of 3^, or a little more [less] than 80 [40] feet : 

 the road must, therefore, be perpendicular to the direction of a line drawn 

 to it from a point 80 [40] feet above the centre of the ring ; and its 

 external circumference must be higher than its internal circumference by 

 one fourth [half] of its breadth. It would, however, be improper to have 

 a road of this form in a manege, since the horse must be taught to perform 

 all his evolutions on a perfect plane. 



There is a general principle of curvilinear motion, which is in itself of 



* Euler on a Rotatory Pendulum, Acta Petr. 1780, pp. 133, 164. 



