AS II XII I BITING THE ROTATION OF T II K K ART II. 29 



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

 j cos $ siir OdO (which corresponds to the integral j y dz of (-4)), 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 t% in the plane of the 

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

 pacing through the zenith at every revolution. Introduce the clement of the 

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

 tii>t of which the plane of revolution moves in azimuth with the angular velocity 

 n sin ?.. The second expressing that there will be an increase of the moment of 

 the quantity of motion about the vertical after a time T proportionate to the area 



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



and at the end of one revolution expresses the area of the circle of radius 7. 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- 

 poM-d very small) the plane will make with the meridian the angle n cos 2.T 7 , and 

 as the quantity of motion in its own plane is t7, its moment referred to a vertical 

 axis will, from zero, have become vP sin (n cos A.T), or, substituting the small arc 

 for its sine, 



vP n cos XT 



%7t 



15ut T, for one revolution, is expressed by - - hence the above becomes 



v 



2n cos 2, n P 



The area of the circle which is generated in the same time is TtP 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 2n cos % 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 n cos Jl t. 



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

 member of equation 4 expresses an angular motion about the complementary axis 

 of which n cos A, 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 n 5 ). If this centri- 

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







of the velocity, indicates no apsidal motion accompanying the decrease of parameters of the orbit. 

 Neither, however, docs it indicate the enlargement of the minor axis 'initially very small) so uni- 

 versally observed in the pendulum experiments. 



