3G LECTURE V. 



tion, theorems of a similar nature have been much extended with great 

 facility. The experiment naturally suggests a familiar proverb, which 

 cautions us against being led away too precipitately by an appearance of 

 brevity and facility. (Plate II. Fig. 25.) 



It has been found that the inconveniences, resulting from the compli- 

 cated apparatus necessary to introduce a cycloidal motion for the pen- 

 dulums of clocks, are more than equivalent to the advantage of perfect 

 isochronism in theory. For since in small cycloidal arcs the curvature is 

 nearly constant, the time of vibration of a simple circular pendulum must 

 be ultimately the same as that of a cycloidal pendulum of the same length ; 

 but in larger arcs, the time must be somewhat greater, because the circular 

 arc falls without the cycloidal, and is less inclined to the horizon at 

 equal distances from the lowest point. This may be shown by a compa- 

 rison of two equal pendulums, vibrating in arcs of different extent : it 

 may also be observed, by an experiment with two simple pendulums of 

 different lengths, that their times of vibration, like those of cycloidal 

 pendulums, are proportional to the square roots of their lengths ; a half 

 second pendulum being only one fourth as long as a pendulum vibrating 

 seconds. 



We have been obliged to suppose the weight, as well as the inertia, of a 

 pendulum, to be referred to one point, since we are not at present prepared 

 to examine the effect of the slight difference between the situations and the 

 velocities of the different parts of the substances, employed in our experi- 

 ments. The nature of rotatory motion requires to be more fully under- 

 stood, before we can attend to the determination of the centres of oscillation 

 of bodies of various figures, that is, of the points in which their whole weight 

 may be supposed to be concentrated, with regard to its effect on the times 

 of their vibrations. 



It is remarkable that the isochronism of pendulums, which is a property 

 so important in its application, may still be preserved, notwithstanding 

 the interference of a constant retarding force, such as the force of friction 

 is in many cases found to be. It has been shown by Newton,* that each 

 complete vibration of a cycloidal pendulum, retarded by a resistance of 

 this nature, will be shorter than the preceding one by a certain constant 

 space, but that it will be performed in the same time. 



There is a great analogy between the vibrations of pendulums, and the 

 revolution of balls suspended from a fixed point. If a body, suspended 

 by a thread, revolve freely in a horizontal circle, the time of revolution 

 will be the same, whenever the height of the point of suspension above the 

 plane of revolution is the same, whatever be the length of the thread. 

 Thus, if a number of balls are fixed to threads, or rather wires, connected 

 to the same point of an axis, which is made to revolve by means of the 

 whirling table, they will so arrange themselves as to remain very nearly in 

 the same horizontal plane. (Plate II. Fig. 26.) 



The time of each revolution of the balls is equal to the time occupied by 

 a double vibration of a pendulum, of which the length is equal to the 

 height of the point of suspension above the plane in which they revolve ; 

 * Principia, Book II. sec. 6. 



