ig Mb R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING 



vis viva 



also be well to note that p* = -. — , even if G do not vanish, and therefore 



(angular njomentum)^ 



that the vis viva cc (angular momentum)^, when the angular momentum has a fixed direction. 



It is needless to carry the solution farther by investigating the path of 07 in the body, the 



position of the principal axes relatively to Oil, 01 at any time, 8iC., since all these questions 



are discussed with the utmost completeness and elegance in M. Poinsot's Theorie de la 



Rotation. 



34. We will conclude this paper by solving the problems of Foucault's Gyroscope as 

 applied to shew the effects of the earth's rotation, as it will furnish a good illustration of the 

 advantages of the methods of this paper in enabling us to form our equations immediately 

 with respect to the most convenient axes. 



The Gyroscope is essentially a body, whose central ellipsoid is an oblate spheroid by 

 reason of its two lesser principal moments being equal, and which is capable of moving freely 

 about its centre of gravity. In this case, if a rapid rotation be communicated to it about 

 its axis of unequal moment, that axis will evidently retain a fixed direction in space however 

 the centre of gravity move, and therefore relatively to a place on the surface of the earth will 

 alter its position just like a telescope, whose axis is always directed to the same star. But 

 there are two other remarkable cases, where the motion about the centre of gravity is partially 

 constrained ; the first, where the axis of rotation is compelled to remain in the plane of the 

 meridian, the second, when it is compelled to remain in the horizontal plane. These we will 

 now consider. 



35. When the polar axis of the central spheroid always lies in the plane of the meridian, 

 let denote the north polar distance of its extremity A. Let OB coincide with the equato- 

 rial axis in the plane of the meridian, and OC with that perpendicular to the same plane, and 

 refer the motion to the axes OA, OB, OC. Now if Q denote the angular velocity of the earth 



d9 

 about its axis, the motions of OA, OB, OC will be due to the velocities Q cos 0, Q sin 6, — 



at 



about them respectively : also the actual velocities of the body about the same axes are 



respectively w, Q, sin d, — -, and the consequent angular momenta Aui,B\lsia d,B-—, where to, — , 

 at at at 



are reckoned positive when the motion about their axes is in the same direction as the earth's 



about its axis. 



