January 29, 1897.] 



SCIENCE. 



173 



second m' / t' times and the new vectors 

 then compounded. 



Center of Mass : To complete the subject 

 of translational motion for an extended 

 body the customary reference is made to 

 the center of mass. 



ROTATION. 



The case of rotation is treated throughout 

 in complete analogy with the foregoing. 

 What was linear velocity constant through- 

 out the body in the above is now angular 

 velocity also constant throughout the bod}' ; 

 what was mass m has become moment of 

 inertia n, and what was force F has become 

 torque T— formally speaking, of course. 

 The results are reached in the usual ele- 

 mentary way. 



The first proposition to be laid down is 

 Lagrange's well-known elementary proof 

 for the composition and resolution of angu- 

 lar velocities. This must be most carefully 

 done ; for if students growl at the sign of a 

 translational velocity they break out in 

 open mutiny at the sign of an angular 

 velocity. Obviously the arrow is again 

 necessary for the complete specification, and 

 I am in the habit of using the sign of Mars 

 ( S) for angular velocities, measuring the 

 arrow from the center of the circle. As a 

 rule, only one of a group of velocities need 

 be so marked. If right-handed relations be 

 postulated (the reverse is the rule in dynam- 

 ics) then an eye looking in the direction 

 of the arrow sees clockwise rotation around 

 it as an axis, with a speed given by the 

 length of the arrow. 



Thus one obtains in succession : 



Angular velocity : 

 (?) 



V-=/ 



reproducing all the propositions (1) to (6) 

 above. Stress must be laid on proposition 

 (5). 



Angular acceleration : Essentially like (8) 

 and (9) above. 



Angular momentum, moment of momentum : 

 If n and ?i' be the moments of inertia the 

 quantities to be compounded are, for in- 

 stance, 



reproducing (10) and (11') above. 



Torque, couple, moment of rotation: If t 

 be sufficiently small, torque is defined in 

 in the most general way by 



where in the first member of the equation 

 the second vectorial term (angular veloc- 

 ity) changes into the first term in the time 

 t for every particle of the mass implied in 

 n. Thus the propositions (12) and (13) 

 are formally reproduced for rotations. In 

 other words, torques, couples, moments are 

 compounded just like forces, and the con- 

 vention involved is the convention made 

 in representing angular velocities. 



I will conclude by giving a few examples, 

 the first of which, FoucauWs Pendulum, is 

 cited merely as a concrete case of (1'). 



Let o) be the earth's angular velocity. 

 Let <p be the latitude of the place of obser- 

 vation. Resolve <y as shown in figure. 



Then <u" rotates the plane of the pendulum 

 around a line in this plane, horizontal for 

 the place. Hence u>" produces no deviation. 

 Obviously <«', the deviating component is 

 <o sin <p. 



In physical meteorology the same result 

 enters fundamentally into the theory of 

 cyclones. For if 2 m Vui be the deviating 

 component of the earth's rotation for a 

 circumpolar body of mass m and velocity F, 

 then the corresponding component for any 



