

Rev. C. S. Lyman on the Pendulum Experiment 409 



relation of the peudulam to that force must also be constant, and 

 Its motion is to be treated, not absokitely or in reference to space, 

 bat precisely as if the earth were at rest, the equal motion of 

 both earth and pendulum having no effect on their mutual rela- 

 tions to each other, .save in the relative change in the horizontal 

 direction of the line of the pendulums vibrations, in consequence 

 of the fact that that direction is entirely independent of the earth's 

 motion, as we have just illustrated. The direction of the plane 

 of vibration depends for its permanency on the inertia of patter, 

 and whatever change may be made in the position of that plane 

 m respect to space, or the heavens, by the earth's rotation, precise- 

 ly the same change of position takes place in the earth itself, and 

 as we have said, the direction of gravity in respect to both re- 

 maining the same, we are only required to consider the relative 

 horizontal change that takes place in the angle made by the plane 

 of vibration with a given line on the earth's surface. 



At the pole the problem is extremely simple^ — the plane of vi- 

 bration remaining constant, and the earth turning under it at the 

 full rate of its angular velocity of rotation, or 15^ per hour. 



-As we recede from the pole, the problem is, to find the true 

 ratio subsisting between the earth's angular velocity of rotation 

 and the latitude of the place, for at the equator both the latitude 

 and motion of the plane are at zero, and no relative angular mo- 

 tion can be exhibited. 



M. Binet, (Comptes Rendus, 1851, Nos. 6, 7), Rev. J. A. 

 Coombe, (Phil. Magazine, No. 7, 1851), and other mathema- 

 ticians, have investigated the problem on the method of resolving 

 the rotary motion of one point on the earth's surface into two, 

 one about the vertical to that point, and the other about an axis at 

 right angles to it and lying in the direction of the meridian. If 

 the motion took place wholly about the latter, which is parallel 

 to the surface of the table over which the pendulum vibrates, the 

 effect would be precisely the same as at the equator, for there 

 this axis coincides with the earth's axis, and in either case there 

 could be no relative motion o^ the plane. It is the other part of 

 the resolved motion therefore, that is alone effectual in giving the 

 table a virtual motion of rotation in its own plane about its center. 

 If the motion were wholly around this axis, the case would be 

 the same as at the pole, for there this vertical axis would coincide 

 with the earth's axis, and the angular motion would be at its 

 maximum. From these considerations equations are formed from 

 which the exact angular change of position of a line on the 

 earth's surface, considered as around the vertical, is shown to be 

 proportional to the siyie of the latitude. M. Binet enters into an 

 analytical investigation, in which the conditions oi^ the motion of 

 the pendulum generally are expressed by certain differential equa- 

 tions; the integration of which conducts him to certain expres- 



