2 REPORT—1857. 
cities 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, quan- 
tities which in general are not independent. But Lagrange introduces, in 
place of the coordinates 7, y,2 of the different particles, any variables or 
(using the term in a general sense) coordinates é, W, ¢,... whatever, deter- 
mining the position of the system at the time ¢: these may be taken to be in- 
dependent, and then if é', W', ¢', .. denote as usual the differential coefficients of 
&,w,,.. with respect to the time, the equations of motion assume the form 
or when ©, ‘¥,®,.. are the partial differential coefficients with respect to 
&, , ¢,... of one and the same function V, then the form 
a aT_ dT aV_ 
didi dé" 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 £, ,,..; and (when such function exists) V, 
or the force function*, is a function depending on the impressed forces and 
expressed in like manner by means of the coordinates &, J, g,..; 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 £,,¢,..., of their differential coefficients é',W/,9!,... 
and of the time ¢; and it is of the second order in regard to the differential 
coefficients é, J’, ¢!,...; and (when such function exists) the force function 
V is a given function of the coordinates é, i), ¢,.. and of the time¢. 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 é', /, g',..; the equations of motiou have in this case an integral 
T+V=A, which is the equation of vis viva, and the problems are distin- 
guished as those in which the principle of vis viva holds good. It is to be 
noticed also that in this case since ¢ does not enter into the differential equa- 
tions, the integral equations will contain ¢ in the form ¢+e, that is, in con- 
nexion with an arbitrary constant ¢ attached to it by addition. 
2. The above-mentioned form is par excellence the Lagrangian form of the 
equations of motion, and the one which has given rise to almost all the ulte- 
rior developments of the theory ; but it is proper just to refer to the form 
in which the equations are in the first instance obtained, and which may be 
called the unreduced form, viz. the equations for the motion of a particle 
whose rectangular coordinates are 2, ¥, z, are 
OB nf OM dM 
Udi, Mena. 
where L=0, M=0,... are the equations of condition connecting the 
coordinates of the different points of the system, and X, p,.. are indeter- 
minate multipliers. 
* 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. 
