508 
laws of the centre of gravity, of areas, and of living force, for 
any such multiple system; and had shown that the corres- 
ponding, but less general, equation of relative motion of a bi- 
nary system, which (by changing a—da’ toa, and m-+m/ to M) 
becomes 
a M 
—_ = —_—_ 2 
dé ay (—a’) ( ) 
can be rigorously integrated by the processes of his new cal- 
culus of quaternions, so as to conduct, with facility, when the 
principles and plan have been caught, to the known laws of 
elliptic, parabolic, or hyperbolic motion of one of the two at- 
tracting bodies about the other. (See the Proceedings of July 
14th and 2lst, 1845, Appendix to Volume III., pp. XXXVil., 
&e.) 
At a subsequent Meeting of the Academy, in December, 
1845, Sir W. Hamilton had shown that the general differential 
equation (1) might be put under this other form: 
2 2 ’ 
0=43.m(8a5$ +7480) +82 Tirana (3) 
and that it might, theoretically, be integrated by an adaptation 
of that ‘* General Method in Dynamics” which he had pre- 
viously published in the Philosophical Transactions of the 
Royal Society of London, for the years 1834 and 1835; and 
which depended on a peculiar combination of the principles of 
variations and partial differentials, already illustrated by him, 
in earlier years, for the case of mathematical optics, in the 
Transactions of this Academy. (See Proceedings of Decem- 
ber 8th, 1845, Appendix already cited, pp. lii., &c.) 
At the same Meeting of December, 1845, Sir W. Hamilton 
assigned the two following rigorous differential equations for 
the internal motions of a system of three bodies, with masses 
m, m’, m’, and with vectors a, B+a, y+a,—thatis, for the 
motions of the two latter of these three bodies (regarded as 
Bey a - 
