396 THE THEORY OF SCREWS. [357- 



Let #j , . . . 6 n represent the co-ordinates of the impulsive screw-chain, and 

 let !,... a n be the co-ordinates of the corresponding instantaneous screw- 

 chain, reference being made to the screw-chains of reference just found. 



Lagrange s equations have the form 



A ( dT .\ -dT- p 

 dt \ddj cfaj 



where T is the kinetic energy, and where PiSc^ denotes the work done 

 against the forces by a twist of amplitude BO.I. 



If Q&quot; denote the intensity of the impulsive wrench, then its component 

 on the first screw of reference is f &quot;0 1} and the work done is 2p 1 &quot;0 l &a l , 

 while, since the chains are co-reciprocal, the work done by Sttj against the 

 components of &&quot; on the other chains of reference is zero, we therefore have 



JV*%$r r fc 



We have also 



T=M(u 1 2 d 1 *+...+u n *d n 2 ), 



when u lt ... u n are certain constants. 



We have, therefore, from Lagrange s equation, 



whence, integrating during the small time t, during which the impulsive 

 force acts, 



in which d is the actual twist velocity about the screw-chain, so that d l = dx l , 

 each being merely the expression for the component of that twist velocity 

 about the screw-chain. 



We hence obtain lt ... n , proportional respectively to 



Pi &quot; Pn 



u 2 



If we make = (11), &c., we have the previous result, 

 PI 



n = (nn)a n . 



358. Conjugate screw-chains of Inertia. 



From the results just obtained, which relate of course only to the chains of 

 reference, we can deduce a very remarkable property connecting instantaneous 

 chains, and impulsive chains in general. Let a. and ft be two instantaneous 

 chains, and let and $ be the two corresponding impulsive chains, then when OL 



