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 — a' too, and m-\-in' to m) 

 becomes 



d^a M 



df2-„^(_a2y 



(2) 



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. (Seethe Proceedings of July 

 14th and 21st, 1845, Appendix to Volume III., pp. xxxvii., 

 &c.) 



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 : 



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. Hi., &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, /3 + a, y-f-a, — that is, for the 

 motions of the two latter of these three bodies (regarded as 



