0 i ae 
ON THEORETICAL DYNAMICS. 5 
= &c.; aud, by a simple combination of the several equations, Lagrange 
deduces expressions for = &ce., in terms of = &C.3 viz.— 
= (a DE+ (Gt aN t+ @g8+ one 
wine aR oa es 
in which, for shortness, 
ae - stands for eee 
The form of the expressions shows at once that (a,b)=—(6,a), so that 
the number of the symbols (a, bd) is in fact fifteen. 
Lagrange proceeds to show, that the differential coefficient with respect 
to ¢ of the expression represented by the symbol (a, 6) vanishes identically ; 
_and it follows, that the coefficients (a, b) are functions of the elements only, 
without the time t. 
The general formule are applied to the problem in hand ; and, in con- 
sequence of the vanishing of several of the coefficients (a, b), it is easy in 
the particular problem to pass from the expressions for — &ec. in terms of 
a 
da 
dt’ 
an elegant system of formule for the variations of the elements of a planet’s 
orbit, in terms of the differentiai coefficients of the disturbing function with 
respect to the elements; but it is not for the present purpose necessary to 
consider the form of the system, or the astronomical consequences deduced 
by means of it. 
7. Lagrange’s memoir of the 13th of March, 1809, ‘On the General 
Theory of the Variation of the Arbitrary Constants in all the Problems of 
Mechanics. —The method of the preceding memoir is here applied to the 
general problem ; the equations of motion are considered under the form 
ddT_ aT ,dV_daQ 
dt dr’ dr dr dr’ 
&e. to those for = &ec. in terms of = &c. The author thus obtains 
a 
where T and V are of the degree of generality considered in the ‘ Mé- 
canique Analytique,’ viz., T is a function of 7, s.. 7", s',.. homogeneous of 
the second order as regards the differential coefficients r', s',... and V is 
a function of 7, s,.. only ; or, rather, the equations are considered in a form 
obtained from the above, by writing T—V=R, viz., in the form 
ddR_dR_da 
dt dr! dr dr’ 
and, as in the preceding memoir, expressions are investigated for the dif- 
; dQ, db 
ferential cvefficients Te &c, in terms of a &c.: these are, as before, of the 
* These are substantially the formule of Lagrange; but I have introduced here and 
elsewhere the very convenient abbreviation, due, I think, to Prof. Donkin, of the symbols 
O(#, 2’) 
(4, 6 y 
