AS EXHIBITING THE ROTATION OF THE EARTH. 29 



To get a clearer idea of what is expressed by the periodic term of (7), — 2 n cos "k 



J cos ^ sill" Odd (which corresponds to the integral J y dz of (-!)), we must revert 



to the latter equation. Conceive the pendulum propelled from a state of rest in 

 the vertical, with a very great angular velocity, denoted by v, in the plane of the 

 meridian. Were the earth motionless, it would continue to whirl in this plane, 

 passing through the zenith at every revolution. Introduce the element of the 

 earth's rotation, and the two terms of equation (4) containing n take effect, by the 

 first of whicli the plane of revolution moves in azimuth with the angular velocity 

 n sin /I. The second expressing that there wUl be an increase of the moment of 

 tlic qiiantity of motion about the vertical after a time T proportionate to tlie area 



generated in that time) y dz. Under these conditions this area is cumulative, 



and at the end of one revolution expresses the area of the circle of radius 1. Let 

 us suppose that the plane of motion turns about a line parallel to the complementary 

 terrestrial axis with an angular velocity n cos ^. At the end of the time T (sup- 

 posed very small) tlie plane will make with the meridian the angle n cos T^T, and 

 as the quantity of motion in its own plane is vl, its moment referred to a vertical 

 axis will, from zero, have become (•/- sin (/} cos XT'), or, substituting the small arc 

 for its sine, 



vT- n cos 7.T 



But T, for one revolution, is expressed by — hence the above becomes 



V 

 2)1 cos ?. 7X P 



The area of the circle which is generated in the same time is nl' and is expressed 



by the integral Cy dz, and it is easy to show for each successive revolution that 



the area J y dz multiplied by 2ii cos X corresponds to an increment of the moment 



of quantity of motion about the vertical which it would receive from a turning of 

 the plane about the complementary axis through the angle u cos 7^ i. 



Hence, for the particular case under consideration, the second term of second 

 member of equation 4 expi-esses an angular motion about the complementary axis 

 of which n cos 7. is the velocity. The resultant of this, and the azimuthal com- 

 ponent, is rotation about an axis parallel to that of the earth, and opposite in 

 direction to the earth's rotation. 



The above theorem can be analytically demonstrated. The quantity N, expres- 

 sive of the tension of the cord, is made up of the centrifugal force due to the pen- 

 dulum's relative angular motion and of the variable component of the force of 

 gravity (neglecting, as we have done, quantities of the order »'-). If this centri- 



fugal force is so great that the component of gravity may be neglected, - will 



c 



of the velocity, indicates no apsiflal motion aceonipanyinp: tlio decrense of parameters of the orbit. 

 Neither, liowever, does it indicate tlie enlargement of the minor axis 'initially very small) so uni- 

 versally observed in tiie penduhiiu experiments. 



