Ball — On Lagrange's Equations. 463 



XLII. — On AN Eleiientaey Pkoof of " Lageange's Equations of 

 Motion in Genekaxized Co-oedinates." By Eobeet S. Ball, 

 LL. D., P. E.. S., Andrews' Professor of Astronomy in the 

 University of Dublin, and Royal Astronomer of Ireland. 



[Read 24th January, 1876.] 



The proofs generally given of these most useful equations depend 

 upon D'Alembert's principle. It is possible that these equations 

 would be more used, even in elementary Dynamical problems, if 

 the method of establishing them were simplified. I can hardly 

 believe that the proof here given is new ; but I have read altogether 

 seven proofs in seven different books, and of these, five depend upon 

 D'Alembert's principle, while the two remaining ones have little 

 or nothing in common with the method I here give. 



Let V denote the potential energy of a Dynamical system, and T 

 the kinetic energy. Let q be one of the n generalized co-ordinates 

 by which the position of the system is specified. 



Suppose the system receive a displacement hq : then the particle 

 of mass 7)1, of which the co-ordinates are x, y, z, receives a displace- 

 ment, of which the components are 



Tq^^' dq^'^' Jq^'^- 

 The forces acting on »i, at ./■, y, s, are 



d'-.c d'^>/ dh 



"'df' ''dl^' "'w 



Hence the quantity of work done, while the displacement hq is 

 made, is 



^ ( dx d~x dy d^y d% dh 



"^ \d^ ' df ^ 1^ ' df '^ d^ ' dt"" 



the symbol 2 extending to all the particles of the system. 



The potential energy of the system is therefore diminished by this 

 amount, whence 



dV fdx d^x dy d-y dz d^z 



~Ji[ ^ ^dq'lf^'dq'dt-'^ d^' di- 



We have also 



^ ^ .idxY (dyy (dz 



