558 The Rev. J. A. Coombe on the Rotation of the Earth. 

 and they move with velocities respectively equal to 



ftj.DN'1 w.OD.sin^^ ; 



ft). AM I or to co.OA .sin 6 V- 

 and (o.dn J co.Od.sindj 



These motions will not be altered if we suppose the earth to 

 be at rest, and the pendulum and table to be carried round it. 

 Let now a motion equal and opposite to that of A be commu- 

 nicated, for the instant, to the pendulum and every point of the 

 line J) Ad. What will be the result ? A will be reduced to rest. 

 And with respect to the point d, and to the points between A 

 and d, since their velocity was originally greater than that of A, 

 they will move with their excess of motion in the direction of 

 the arrow ; while with regard to D, and points between it and 

 A, their velocity being originally just as much in defect of A, 

 they will be made to move round A in the direction of the arrow 

 with velocities exactly coiTesponding to those between A and d. 

 The effect will therefore be a rotatory motion round A ; and we 

 proceed to prove, that, for the instant, this motion will be in the 

 plane of the table. 



We suppose that for an instant all the points in the plane of 

 the table are moving in the tangent plane to the surface at A. 

 Let df be a tangent to the circle dff ; this will be the direction 

 of d'& motion ; 



.'. dfis pei'pendicular to dn: 



and dn being the common intersection of the planes of the meri- 

 dian and that of the parallel circle dff, it follows that df is per- 

 pendicular to the plane Vdn, and therefore perpendicular to Ad. 

 Hence d/is a. tangent to the plane of the table ; and so of the 

 other points. Hence, for the instant, every point in the table is 

 turning in its own plane about A fixed with a certain angular 

 velocity. To ascertain this, we have 



vel. of A- vel. of D = ftj.(OA-OD).sin^ 

 = a).AD.sind, 

 and vel. of d— vel. of A = ft) .A</.sin 0. 



And so the motion of every point of the table will be propor- 

 tional to its distance from A ; and the tabk will rotate about A 

 with angular velocity 



= (o.sm6, 



= (o . sin X,, 



as is evident from the figure. 



Of this motion the plane of vibration does not participate, for 

 reasons before assigned. And with regai'd to any motion which 



