ON THEORETICAL DYNAMICS, 13 
26. The first memoir contains applications of the method to the problem 
of two bodies, and the problem of three or more bodies, and researches in 
reference to the approximate integration of the equations of motion by the 
separation of the function V into two parts, one of them depending on the 
principal forces, the other on the disturbing forces. The method, or one of 
the methods, given for this purpose, involves the consideration of the varia- 
tion of the arbitrary constants, but it is not easy to single out any precise 
results, or explain their relation to the results of Lagrange and Poisson. The 
like remark applies to the investigations contained in Nos. 7 to 12 of the 
second memoir, but it is important to consider the theory described in the 
heading of No. 13, as “giving formule for the variation of elements more 
analogous to those already known.” The function H is considered as con- 
sisting of two parts, one of them being treated as a disturbing function ; the 
equations of motion assume therefore the form 
dy_dtl, dy da__dil_ay 
dt da'dza dt dn dn 
(I have written H, Y instead of the author’s H,,H,). The terms involving 
 Y are in the first instance neglected, and it is assumed that the integrals 
of the resulting equations are presented in the form adopted by Poisson, 
viz., the constants of integration a, 6, &c., are considered as given in terms 
of ¢, and of the two sets of variables n,.. and a,..; the integrals are then 
extended to the complete equations by the method of the variation of the 
elements. The resulting expressions are the same in form as those of 
Poisson, viz. :— 
da dy 
Gn bat ae 
where 
O(a, 5) 
(a, yaaa) 7 
(a,b) da db db da 
(7,7) dnda dnda 
if, for shortness, 
and conversely the values of ae &ce. in terms of = &e. might have been 
a 
exhibited in a form such as that of Lagrange. The expressions (a, 5), con- 
sidered as functional symbols, have the same meanings as in the theories of 
Poisson and Lagrange ; and, as in these theories, the differential coefficient 
of (a, 6) with respect to the time, vanishes, or (a, 6) is a function of the 
elements only. 
27. It is to be observed that the disturbing function Y is not necessarily 
in the same problem identical with the disturbing function Q of Lagrange 
and Poisson (indeed, in any problem, the separation of the forces into prin- 
cipal forces and disturbing forces is an arbitrary one). Sir W. R. Hamil- 
_ ton, in the second memoir, gives a very beautiful application of his theory 
to the problem of three or,more bodies, which has the peculiar advantage of 
making the motion of all the bodies depend upon one and the same disturbing 
_ function*. This disturbing function contains (as in the last-mentioned 
* Lagrange has given formule for the determination of the motion of three or more 
bodies referred to their common centre of gravity by means of one and the same disturbing 
function. In Sir W. R. Hamilton’s theory there is one central body to which all the others 
