have in this case an integral T + V = h, which is the equation of vis viva, and the 
problems are distinguished as those in which the principle of vis viva holds good. It 
is to be noticed also that in this case since t does not enter into the differential 
equations, the integral equations will contain t in the form t + c, that is, in connexion 
with an arbitrary constant c 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 ulterior 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 x, y, z, are 
d~x „ dL dM 
m^ = X+ \—+fi -5— + .. • 
where L = 0, M = 0, ... are the equations of condition connecting the coordinates of the 
different points of the system, and y, ... are indeterminate multipliers. 
8. The idea of a force function seems to have originated in the problems of 
physical astronomy. Lagrange, in a memoir “ On the Secular Equation of the Moon,” 
crowned by the French Academy of Sciences in the year 1774, expressed the attractive 
forces, decomposed in the directions of the axes of coordinates, by the partial differential 
coefficients of one and the same function with respect to these coordinates. And it 
was in these problems natural to distinguish the forces into principal and disturbing 
forces, and thence to separate the force function into two parts, a principal force 
function and a disturbing function. The problems of physical astronomy led also to 
the idea of the variation of the arbitrary constants of a mechanical problem. For as 
a fact of observation the planets move in ellipses the elements of which are slowly 
varying; the motion in a fixed ellipse was accounted for by the principal force, the 
attraction of the sun ; the effect of the disturbing force is to produce a continual 
variation of the elements of such elliptic orbit. Euler, in a memoir published in 1740 
in the Memoirs of the Academy of Berlin for that year, obtained differential equations 
of the first order for two of the elements, viz. the inclination and the longitude of 
the node, by making the arbitrary constants which express these elements in the fixed 
orbit to vary : this seems to be the first attempt at the method of the variation of 
the arbitrary constants. Euler afterwards treated the subject in a more complete 
manner, and the method is also made use of by Lagrange in his “Memoir on the 
Perturbations of the Planets ” in the Berlin Memoirs for 1781, 1782, 1783, and by 
Laplace in the Mécanique Céleste, t. 1. 1799. The method in its original form 
seeks for the expressions of the variations of the elements in terms of the differential 
coefficients of the disturbing function with respect to the coordinates. As regards one 
element, the longitude of the epoch, such expression (at least in a finite form) was 
first obtained by Poisson in his memoir of 1808, to be spoken of presently; but I 
am not able to refer to any place where such expressions in their best form are even 
now to be found ; the question seems to have been unduly passed over in consequence 
of the new form immediately afterwards assumed by the method. It was very early