406 THE THEORY OF SCREWS. [366- 



irom Lagrange s equations, 



d (dT\ dT 



~ = ^W.T?] . 



dt \d0J dd, w - 



dt\de 



These equations admit of a transformation by the aid of the identity 



e^- f dT -o 

 *ap&quot; n de n - 



Differentiating this equation by & lt we find 



but 



dT * _ , 



** 



= Q _ _ , 



dt \dej ~ J dd? 2 de.de, n de, de n * de^ei + n de~de~f 



whence, by substitution 



AfdT.} = 0fT d * T dT 



dt\dej ~ l de^&quot; n 



Hence when screw-chain co-ordinates are employed Lagrange s equations 

 may be written in the form 



ifT d T d T 



* 



367. Generalization of the Eulerian Equations. 



The equations just written can be further simplified by appropriate 

 choice of the screw-chains of reference. We have already assumed the 

 screw-chains of reference to be co-reciprocal. If, however, we select that 

 particular group which forms the principal screw-chains of inertia ( 357), 

 then every pair are conjugate screw-chains of inertia besides being reciprocal. 

 In this case T takes the form 



