195] REPORT ON THE RECENT PROGRESS OF THEORETICAL DYNAMICS. 157 
mentioned special problems, I shall have frequent occasion to allude to: I mean the 
problem of the variation of the elements of a planet’s orbit, which has a close historical 
connexion with the general theories which form the subject of this report. The so-called 
ideal coordinates of Hansen, and the principles of his method of integration in the 
planetary and lunar theories, have a bearing on the general subject, and might have 
been considered in the present report; but on the whole I have considered it better 
not to do so. 
1. Lagrange, Mécanique Analytique, 1788.—The equations of motion are obtained, 
as before mentioned, by means of the principle of virtual velocities and d’Alembert’s 
principle. In their original forms they involve the coordinates x, y, z of the different 
particles m or dm of the system, quantities which in general are not independent. 
But Lagrange introduces, in place of the coordinates x, y, z of the different particles, 
any variables or (using the term in a general sense) coordinates £, <£,... whatever, 
determining the position of the system at the time t: these may be taken to be 
independent, and then if \Jr’, </>',... denote as usual the differential coefficients of 
f, yjr, cf),... with respect to the time, the equations of motion assume the form 
d clT dT „ n 
dt dÇ dÇ + ~ ’ 
or when B, M/“, <E>,... are the partial differential coefficients with respect to £, yfr> ••• 
of one and the same function V, then the form 
c±dT_dT dV 
dt d£ d^ + d£ 
In these equations, T, or the vis viva function, is the vis viva of the system, or sum 
of all the elements each into the half square of its velocity, expressed by means of 
the coordinates £, фг, ф, ... ; and (when such function exists) V, or the force function( x ), 
is a function depending on the impressed forces and expressed in like manner by 
means of the coordinates £, фг, ф,... ; the two functions T and V are given functions, 
by means of which the equations of motion for the particular problem in hand are 
completely expressed. In any dynamical problem whatever, the vis viva function T is 
a given function of the coordinates £, фг, ф, ..., of their differential coefficients 
фг, ф',... and of the time t; and it is of the second order in regard to the 
differential coefficients фг', ф',... ; and (when such function exists) the force function 
F is a given function of the coordinates £, фг,ф,... and of the time t. This is the 
most general form of the functions T, V, as they occur in dynamical problems, but 
in an extensive class of such problems the forms are less general, viz. T and V are 
each of them independent of the time, and T is a homogeneous function of the second 
order in regard to the differential coefficients f', фг', ф', ... ; the equations of motion 
1 The sign attributed to V is that of the Mécanique Analytique, but it would be better to write 
V= -U, and to call U (instead of V) the force function.