6 REPORT—1857. 
oe ee 
form i =(a, b)- +, &e, 
where (a, 5), &c., are in the body of the memoir obtained under a some- 
what complicated form, and this complicates also the demonstration which 
is there given of the theorem, that (a, b), &c., are functions of the elements 
only, without the time t; but in the addition (published as part of the me- 
moir, and without a separate date) and in the supplement the investigation 
is simplified, and the true form of the functions (a, 4) obtained, viz., writing 
dT 
fn ee then . 
(7p) , O(s, ¢) 
Cae atest jak meh) 
if, for shortness, 
; 0(7; p) p)_ar dp _dp dr c 
0(a4,6) dadb dad ~ 
dT dT 
The representation of a? as? &c. by single letters 7s made by Lagrange 
in the addition, No. 26 (Lagrange writes Sa == T", &c.), but quite 
incidentally in that number only, for the sake of the formula just stated: I 
have noticed this, as the step is an important one. 
8. It is proper to remark that, in order to prove that the expressions 
(a, 6) &c., are independent of the time, Lagrange, instead of considering 
the differential coefficients of each of these functions: separately, establishes 
a general equation (see Nos. 25, 34, 35 of the Addition, and also the Sup- 
plement) is ‘ Pa 
d ' 
5(a” aaa qt) = 
where, if Aa, Ab,.. denote any arbitrary increments whatever of the con- 
stants of integration a, b,.. then Ar, &c., are the corresponding increments 
of the coordinates 7, &c.; this is, in fact, a grouping together of several 
distinct equations by means of arbitrary multipliers, and it is extremely 
elegant as a method of demonstration, and has been employed as well by 
Lagrange, here and elsewhere, as by others who have written on the 
subject ; but I think the meaning of the formule is best seen when the 
component equations of the group are separately exhibited, and in the 
citation of formule I have therefore usually followed this course. Lagrange 
gives also an equation which is in fact a condensed form of the preceding 
: dQ ii int : , 
expression for da’ but which it is proper to mention, viz. :— 
dQ, dr dR d dR 
== 3s 7 oy Sees 
da da°dr' rs vai dr’ 
; dR d dRda, d dR db 
i , in the f Na vons=a5 si SE aes Mai eied eee 
In fact, in the formula a stands for (5 ey ae ah di el at and 
dr da , dr db Rossi se Bg 
or for & a sie at its and, on substituting these values, the identity 
of the two expressions is seen without difficulty. 
9. Lagrange remarks, that in the case where the condition of wis viva 
holds good, then if @ be the constant of vis viva (T+v=a), and e the 
constant attached by addition to the time, then oF, which, he observes, 
