1886.] Mr M*Connel, On Lagrange’s Equations of Motion. 25 
(4) On Lagrange’s Equations of Motion. By James C, 
M° ConneEt, M.A. 
The following proof of Lagrange’s equations of motion is founded 
on that given by Lord Rayleigh in The Theory of Sound. It is 
simplified by supposing the arbitrary displacements of the system 
not to vary with the time. This alteration brings more clearly 
into view the part really played by the last term in the final 
equations of motion. We shall suppose for simplicity that the 
energy of the system is entirely kinetic. 
Let the generalised coordinates be ,, ,..., which are inde- 
pendent, and sufficient to determine the configuration. Let the 
generalised components of force be V,, V,..., and let XYZ be the 
forces acting on any particle of mass m at the point 2yz. Then if p, 
be a small ‘arbitrary increment of yw, and aPy the corresponding 
increments of wyz, we have by virtual velocities 
Wo. => (Xa de VO Fy aera eae: (1). 
dic dy dz 
Now a dap, Pv aan ons TS app ee 
Le B(x at oe +2ae Liha: (2). 
This equation is a simple statement of the process by which we 
can obtain VW, from the X, Y and Z of all the particles. 
Now since 
da 
dy Vy oF dp, vy, 
di _ de 
dy, ap, 
: 2 ( en —) 
dyp, dnp, dyp, dnp 
= 3 (ma oF aan my —— dy + mz ral BG Seta (8). 
vy, dyp, dN, 
Thus aE can be obtained from the momenta of the particles 
by exactly the same process as that by which VY, was obtained from 
the forces acting on the particles. 
