174 



SCIENCE. 



[N. S. Vol. V. No. 109. 



latitude <p is 2 m V<" siu ip, quite independ- 

 ent of the azimuth of V. 



Again, if in figure (15) w and w^ be re- 

 placed by linear velocities, one easily ob- 

 tains by (8) the expression for accleration 

 towards a center, etc. 



Precession : In instruments like tops, gyro- 

 scopes, etc., the mechanism (supposed fric- 

 tionless) is such as to exclude all interference 

 from without, with the magnitude of the 

 angular velocity lo of the top around its 

 axis. This constructive condition is essen- 

 tial. Hence, if the axis changes position, and 



if for brevity we suppose the tail of the ar- 

 row (u to remain frictionlessly at C, then the 

 locus of the point of the arrow must be the 

 surface of a sphere of radius w. Let <« change 

 in position to <Uj, let the axis to which the 

 change of angular velocity w' (in figure 15) 

 corresponds, pass through G and necessarily 

 rotate around it in a horizontal plane. This 

 is clearly the case with the axis of gravita- 

 tional torque in the precessional motion of 

 a top or gyroscope. Then must w' also lie 

 in a horizontal plane, and the locus of w is 

 the surface of a circular cone with its axis 

 vertical and its vertex at C. If u>' is im- 

 parted in unit of time <«' is the mean an- 

 gular acceleration due to the gravitational 

 torque and therefore equal to T/n by (12'). 

 But the inclination of w to the horizontal 

 has just been shown to be constant (cone), 

 wherefore gravitational torque is constant 

 and (u' is constant. Hence the precessional 

 motion is uniform rotation around the ver- 

 tical axis of the fixed cone ; for from one 

 point of view to' is the total change of an- 

 gular velocity due to gi-avitational torque, 

 and from another point of view, w'/"', con- 

 stant for the reasons specified, is propor- 

 tional to the uniform angular velocity of 



precession (see figure). If gravitational 

 torque is withdrawn, as in a balanced gyro- 

 scope, (/j'/(«=:0 and precession ceases. If 

 o) gradually decreases (friction), w' will 

 subtend a relatively greater angle, or pre- 

 cessional motion will be accelerated, even 

 when the axis of <« is not lowered. In the 

 latter case the result is accentuated, for 

 gravitational torque is increased. 



Again, suppose gimbals of a gyroscope 

 forcibly rotated around a vertical axis. In 

 Figure 16 let the angular velocity w be thus 



imparted in unit of time. Let <«j and ut^ 

 be the positions of the top axis and its an- 

 gular velocities before and after the inter- 

 ference. Resolve <u into components u>' and 

 <u" respectively at right angles and parallel 

 to (Uj. Then m" would rotate the top axis 

 if it were not frictionlessly mounted. It 

 actually rotates the gimbals only. There- 

 fore u)^ = <«2 in length, as is otherwise evi- 

 dent. Thus <u' is the total effective change 

 of angular velocity, and in virtue of this w^ 

 passes to <«, and the extremity of the top 

 axis rises, describing a circle in a vertical 

 plane. If <« is imparted in a contrary di- 

 rection the motion of u)^ will be reversed. 

 The top rolling on a blunt point belongs 

 here. 



Finally, if the top axis is forcibly rotated 

 back and forth over a small angle around 

 the horizontal axis of gravitational torque, 

 similar considerations will lead to a better 

 explanation of the curves drawn by a top 

 on an inclined plane than I gave in a pre- 

 ceding article. The periodic changes of 

 torque correspond to the rolling of the top 

 up and down the inclined plane. 



I have been tempted to enter somewhat 

 at length into this most important subject. 



