160 
REPORT ON THE RECENT PROGRESS OF THEORETICAL DYNAMICS. [195 
of the integrals, and the memoir may be considered as the foundation of the general 
theory. The equations of motion are considered under the form, 
d?x 
1 + m 
dii 
dt 2 
X ~ dx’ 
d-y 
1 + m 
dii 
dt 2 
II 
©j 
^ i 
d-z 
1 + m 
dii 
dt 2 
rp 3 
Z dz ■ 
and it is assumed that the terms in il being neglected, the problem is completely 
solved, viz., that the three coordinates, x, y, z, and their differential coefficients, 
x, y', z, are each of them given as functions of t, and of the constants of integration 
a, b, c, f, g, h; the disturbing function il is consequently also given as a function 
of t, and of the arbitrary constants. The velocities are assumed to be the same as 
in the undisturbed orbit. This gives the conditions 
£ = o' ^ +■ 
r. C JL 
- d X + 5T 
0\ 
J, ?.. 
8x = 0, 8y = 0, 
and then the equations of motion give 
/ X * 
8z = 0; 
é_x m 
9<V (Mr 
g dx _ dii g dy _ dii g dz _ dii 
dt dx ’ dt dy ’ dt dz ’ 
equations in which 8x, &c. denote the variations of x, &c., arising from the variations 
of the arbitrary constants, viz., 8x = ^ 8a + ^ 8b +, &c. The differential coefficients 
^ , &c., can of course be expressed by means of ^ , &c.; and, by a simple combi 
nation of the several equations, Lagrange deduces expressions for &c., in terms of 
da 
dt 
, &c. ; viz. 
dii , ,.db . N dc , df . . dq , 7x dh 
da =(a ’ 6) S + ( “’ c) dt + (a ’ f) Ì + {a ’ g) dt +(a ’ h) df 
where (*) 
in which, for shortness, 
(a - d I d y") I d Z ') 
v ’ ' 9 (a, byd(a, byd(a, 6)’ 
9 (x, x') J _ dx dx' dx' dx 
ndTb) stands for dcTdb--¿kTdb- 
1 These are substantially the formulas of Lagrange; but I have introduced here and elsewhere the very 
3’ (iC 
convenient abbreviation, due, I think, to Prof. Donkin, of the symbols , , ’ ,. 
J O (a, b)