Ball — On the Application of Lafj rangers Equations. 213 



XXXV. J^OXE OJr THE ApPLICATIOJT of LaGRAXGe's EQTJATIOJfS OF 



Motion to Pkoblems in" the Dynamics of a Eigid Body. By 

 EoBEET S. Ball, LL. D., ¥. B. S., Andrew Prof essor of Astronomy 

 in the UmTersity of Dublin, and Boyal Astronomer of Ireland. 



[Read February 24, 1879]. 



The problem to which I wish to direct attention occurs in the Theory 

 of Screws, and is thus expressed in the language of that Theory. 



A quiescent rigid body has freedom of the ?*"' order : being given 

 the co-ordinates of an impulsive wrench, it is required to find the co- 

 ordinates of the corresponding instantaneous screw. 



The solution of this problem is given in the Theory of Screws, 

 p. 60. The method there adopted is quite different from that now 

 communicated, which is founded on Jj^.^xdJi^e' s Equations of Motion in 

 Generalized Co-ordinates. 



Without any loss of generality we may assume that the impulsive 

 wrench is on a screw which belongs to the screw system, defining the 

 freedom of the body ; for, owing to the reactions of the constraints, 

 one screw (but only one) can always be found in the screw system, a 

 wrench on which would produce the same effect as a wrench on a 

 screw otherwise placed. 



Under these circumstances, let ti, &c., ^,„ represent the co-ordinates 

 of the impulsive screw, and 6^, &c., 6,^, be the co-ordinates of the cor- 

 responding instantaneous screw, reference being made as usual to the 

 principal screws of inertia. 



Lagrange's equations are typified by 



d(dT\_cIT^_ 



where Tis the kinetic energy, and where PiWi denotes the work done 

 in a twist Wi against the forces. 



If C" represent the intensity of the impulsive wrench, then 



Pi = 2p,CZ 



T = M{u,%' + &c. + u,^e,^), 



where pi, p,„ &c., are the pitches of the principal screws of inertia, 



112 



