268, 269] MOTION OF A CHAIN. 305 



of inextensibility. In the first place the velocity of P' has com- 

 ponents 



du 7 dv , 



u + =- as, v + ^~ ds, 



ds ds 



along the tangent and normal at P', and these lines make angles 

 p ds with the tangent and normal at P. Now the velocity of P' 



Fig. 83. 



relative to P must ultimately be at right angles to PP, and 

 therefore we have 



, 9l * ,7 \ fi$ 1 \ ( , dV J ^\ ' fi$ j \ A 



u + 5- ds} cos { ^ ds} u (v + ^ ds I sin I ^- as = 0, 

 85 J \ds J \ ds J \ds ) 



da v 

 or - --- =0 ........................... (1). 



ds p 



Again, since PP' is a constant length and turns in the plane 



with angular velocity -~ , the velocity of P' relative to P resolved 

 ot 



at right angles to PP' is ultimately ds ~ , and thus we have 



dv , \ /d6 7 \ / 9^ , \ . /3<f> 7 \ ,90 



v + - 5 1 cos [~ as] + {u + -z- d8]8Ui i-f ds v = ds , 

 85 / \8 / V ds J \ds J dt 



dv u d(f> 



or ^- + - = ^7 ........................... (2). 



ds p dt 



The equations (1) and (2) are the kinematic conditions of inexten- 

 sibility of the chain. 



To form the equations of motion of the chain under any forces 

 we have to observe that u t v, <f> are functions of two independent 

 variables s and t, and that the centre of inertia of every element of 

 the chain moves under the action of the resultant of the tensions 

 at its ends and of the bodily forces exerted upon it. 



L. 20 



