of which we can perfectly define any infinitesimal change in the configuration of the system; and let
where are to be determined by the change in the configuration in the interval of time ; and let
Also let
It is evident that can be expressed in terms of , etc., , etc., and the quantities which express the configuration of the system, and that (since is used to denote a variation which does not affect the configuration or the velocities),
Moreover, since the quantities in the general formula are entirely determined by the configuration of the system
where denotes the ratio of simultaneous values of and , when , etc. are equal to zero, and , etc are to be interpreted on the same principle. Multiplying by , and taking the sum with respect to the several forces, we have
where
If we differentiate with respect to , and take the variation denoted by , we obtain
The general formula (12) is thus reduced to the form
(26)
If the forces have a potential , we may write
(27)
where denotes the ratio of and when , etc. have the value zero, and the analogous expressions are to be interpreted on the same principle.