characteristic Quantities occurring in Central Motions. 9 



gular motion of the radius vector, and the to-and-fro motion of 

 the point within the radius vector. 



The angular motion of the radius vector we will name motion 

 of rotation, and understand by period of rotation the time during 

 which the radius vector runs through the entire angular space 

 2tt. The motion of the point within the radius vector we will 

 name radial motion of vibration, or, briefly, motion of vibration, 

 and choose the name period of vibration for the time during which 

 a vibration to and fro takes place. The period of a rotation may 

 be denoted by i, and the period of a vibration by i v 



The motion of vibration can be treated as quite independent 

 of the motion of rotation, if we introduce the centrifugal as a 

 special force. Let 6 denote the angle which the radius vector 

 forms, at the time /, with a fixed right line in the plane of rota- 

 tion, so that -=- will denote the angular velocity of the radius 



vector ; then the centrifugal force to which the rotation gives 

 rise will be represented by the product 



(dd\* 



mr (dt) ' 



Now, as for the motion of a point about a fixed centre of 

 attraction the proposition is valid that the radius vector describes 

 equal spaces in equal times, we have the equation 



+£•* ....... (7) 



in which c is a constant ; and from this we immediately obtain, 

 further, 



mr 





The centrifugal force represented by this expression we will now 

 regard as a force of repulsion exerted from the centre, and to be 

 added to the force represented by F(r), which is actually exerted 

 from the centre. Then we obtain for the motion of vibration 

 the following differential equation, 



with the aid of which we can treat the vibration as a motion 

 subsisting by itself. To this motion thus regarded, both my 

 equations are found at once applicable. 



Equation (2) gives, if we substitute r for x, and — F'(r) -f mc 2 -g- 



