333 
1908-9.] On Lagrange’s Equations of Motion 
are fulfilled, (16) reduces to 
0T _0T 
dt dq Y dq l 
(18). 
12. Equations (17) form one set of the conditions which make equations 
(9) the derivatives of a set of finite equations ; and if all these conditions 
are fulfilled, the equations of motion take the usual form typified by (18). 
The fulfilment of each set involves the validity of a corresponding equation 
of motion in the usual form. 
It has been remarked by Appell (Mecaniqne Rationelle, t. ii., art. 462) 
that if one set, say (17), of these conditions holds, we can write (using our 
present notation) 
Su SI “H CL.^Scjr, “j - ^-g^^g 
8v = 8G + f3 2 8q 2 + f3 s 8q s + . . . j . . (19) : 
where <£F, <SG, are perfect differentials. 
If, then, we notice when the fundamental equations are written down 
that the terms corresponding to any co-ordinate are thus expressed, we 
know that the equation of motion corresponding to that co-ordinate can be 
found by the ordinary Lagrangian process. 
13. In what follows we shall, for brevity, adopt the notation 
<^> 2 1 = -f- b^v -4- . . . . \ 
<£ 2 T = d 2 u + b 2 v + .... . . . . (20). 
If the form of T be modified in any way, for example, by the sub- 
stitution of the values of s v s 2 , ... . in terms of the momenta dT/ds lt 
0T/0s 2 , . . . . , and the other velocities q v q 2 , .... , then if T' be the 
modified form of T, < p X T', cp 2 T', .... will be understood to denote the 
operations indicated in (20), but performed with the new values which are 
then given to the coefficients of q v q 2 , .... , q k . 
14. It follows from what has been stated that, as has already been 
pointed out, the operations indicated in (20) cannot be performed without 
reference to the fundamental equations from which that expression has 
been derived. For example, two terms in T might be A Ad 2 -f- JBi/A 
These might be derived either from ii= JA.6, v= JByjs, or from 
u = J A sin 6.6+ VL cos 6 .\js, v= JA cos 6.6— ^JB sin 6 . \[r. The former 
mode of derivation would satisfy the conditions of integrability so far 
as these terms are concerned, the second would not. It is possible, in 
fact, to specify two distinct cases of motion which have precisely the same 
expressions for the kinetic and potential energies, but which have not the 
