116 Proceedings of the Royal Society of Edinburgh. [Sess. 
non-modified co-ordinates is concerned it entirely replaces the ordinary 
Lagrangian function. 
4. It is of importance to notice that the process employed above to render 
the velocities and co-ordinates independent as regards the initial variational 
problem need not be carried out in its entirety, it being only necessary to 
carry it to the extent of rendering it valid for the co-ordinates to be 
modified. Thus, for example, for the purposes of the last paragraph, 
we need only consider the variation of the integral 
ft \~ m m 
l t ^ ~~ + tPr 
i l_ r=l r— l 
the variation with respect to all but the first m co-ordinates being effected 
in the usual manner. 
This remark leads us to our next point. The complete equations of 
motion are equivalent to the conditions for the vanishing of the variation 
of this integral taken as if the velocities (or their momenta) and the co- 
ordinates are independent if they are represented in the sum 2- Now, in 
the special case when the momenta are all constant in time this integral 
may be replaced by the integral 
j f '( L - jtprfr ft = jf X'dt 
for the outstanding terms 
reduce to a set of constants depending only on the initial and final configu- 
rations, and cannot therefore contribute anything to the general expression 
for the variation. This is the result derived by Larmor, that it is in this 
special case that the integral 
possesses the minimum property usually associated with the Hamiltonian 
integral. 
Larmor’s result has a still more general significance, for in all cases, 
whether the momenta in the modified co-ordinates are constant or not, the 
variation of the integral 
P 2 L 'dt 
Jti 
with respect to the non-modified co-ordinates leads to the proper equations 
for the motion in those co-ordinates, for the remaining part of the complete 
