329 
1908-9.] On Lagrange’s Equations of Motion. 
pressing variations Sq k+1 , . . . Sq k+m of m other parameters connected by m 
kinematical equations with the variations of the parameters q v q 2 , . . . , q k 
are taken account of. We shall also suppose for the present that the kine- 
matical equations do not involve the time t explicitly. 
6. From (2) the values of x, y , z can be found, and from these again 
the values of x, y, z. The latter include, besides terms in q v q 2 , ... , q k) 
also terms involving the time-rates of variation of the a, b, c coefficients. 
If, then, we substitute in the expression -JS{m(x 2 + y 2 + £ 2 )}, we get a 
transformed expression of which part, which we may call S, is an explicit 
function of q v q 2 , . . . , q k . If then we write 
2KX + b i Y + c,-Z) = Q*, 
the generalised force according to the usual specification, we obtain 
P|-Qi, P = Q 2 . (3), 
a^i dq 2 
which are Appell’s equations. 
7. Consider now the equations of motion 
mx = X, my = Y , mz = Z . . . (4), 
of a free particle, and from the equations of this form for the particles of 
the system construct 
S{m(a 1 aj + byj + c-^z)} = + IqY + c Y Z)] 
2 {m(a 2 x + b 2 y + c 2 z ) } = %(a 2 X + b 2 Y + c 2 Z) l . . . (5). 
The quantities on the right-hand sides in (5) are the generalised forces 
Qp Q 2 , .... of the Lagrangian equations. 
It will be observed that since any Q is the coefficient of Sq in the 
expression Q Sq for the work done in a possible variation of the parameter 
q, Q does not include any of what may be called the non-active forces — 
that is, forces such as those due to guides and constraints which are 
invariable. We have then to consider the equation 
1{m(ax + bij + cz)} = Q ..... (6). 
Lord Kelvin’s process was as follows : — Writing dx/dq, dy/dq, dz/dq instead 
of a, b, c, for the system was tacitly supposed to be holonomous, he obtained 
(in a slightly different notation) 
.me d ( .dx\ . d dx 
dq dt\ dqj dt dq 
and then proved that 
dx dx d dx dx 
dq dq ’ dt dq dq ’ 
