ON THEORETICAL DYNAMICS. f 
is an equation remarkable as well from its simplicity and generality as 
because it can be obtained @ priori, independently of the variations of the 
other arbitrary constants: this is obviously the generalisation of the expres- 
‘sion for the variation of the mean distance of a planet. 
10. The consideration of Lagrange’s function (a, 6) originated, as appears 
from what has preceded, in the theory of the variation of the elements; but 
it is to be noticed, that the function (a,5) is altogether independent of the 
disturbing function, and the fundamental theorem that (a, 6) is a function 
of the elements only, without the time, is a property of the undisturbed 
equations of motion. The like remark applies to Poisson’s function (a, 6), 
in the memoir next spoken of. 
11. Poisson’s memoir of the 16th of October, 1809. The formule of 
this memoir are, so to speak, the reciprocals of those of Lagrange. The 
relations between the differential coefficients = &c., of the disturbing func- 
a 
; airs d 
tion and the variations a &c., of the elements, depend with Lagrange, 
upon expressions for the coordinates and their differential coefficients in 
terms of the time and the elements; with Poisson, on expressions for the 
elements in terms of the time, and of the coordinates and their differential 
coefficients. The distinction is far more important than would at first 
sight appear, and the theory of Poisson gives rise to developments which 
seem to have nothing corresponding to them in the theory of Lagrange. 
The reason is as follows: when the system of differential equations is com- 
pletely integrated, it is of course the same thing whether we have the 
integral equations in the form made use of by Lagrange, or in that by 
Poisson, the two systems are precisely equivalent the one to the other ; but 
when the equations are not completely integrated, suppose, for instance, we 
have an expression for one of the coordinates in terms of the time and the 
elements, it is impossible to judge whether this is or is not one of the inte- 
gral equations ; the differential equations are not satisfied by means of this 
equation alone, but only by-this equation with the assistance of the other 
integral equations. On the other hand, when we have an expression for 
one of the constants of integration in terms of the time, and of the co- 
ordinates and their differential coefficients, it is possible, by mere substi- 
tution in the differential equations, and without the knowledge of any other 
integral equations, to see that the differential equations are satisfied, and 
that the assumed expression is, in fact, one of the system of integral 
equations. An expression of the form just referred to, viz.,c=¢ (Ganyice 
z',y'...), where the right-hand side does not contain any of the arbitrary 
constants, may, with great propriety, be termed an “ integral,” as distin- 
guished from an integral equation, in which the constants and variables 
may enter in any conceivable manner ; it is convenient also to speak of such 
equation simply as the integral e. 
12. Returning now to the consideration of Poisson’s memoir, the equa- 
tions of motion are considered under the same form as by Lagrange, viz., 
putting T—V=R under the form 
dt dy’ dy ap’ 
but Poisson writes : 
dR 
— =S, ee 
dg 
thus, in effect, introducing a new set of variables, s,.. equal in number to 
