No. V. 
December 8, 1845. (See page 150). 
Tue following is the communication made by Sir William R. 
Hamilton on some additional applications of his theory of 
algebraic quaternions. 
It had been shown to the Academy, at one of their meet- 
ings* in the last summer, that the differential equations of 
motion of a system of bodies attracting each other according 
to Newton’s law, might be expressed by the formula : 
da m + Am 
dé? — Aa ¥( — Aa’) (1) 
in which a is the vector of any one such body, or of any ele- 
mentary portion of a body, regarded as a material point, and 
referred to an arbitrary origin; m the constant called its mass ; 
a + Aa, and m + Am, the vector and the mass of another 
point or body of the system; = the sign of summation, rela- 
tively to all such other bodies, or attracting elements of the 
system ; and d the characteristic of differentiation, performed 
with respect to ¢ the time. ¢ 
If we confine ourselves for a moment to the consideration. 
of two bodies, m and m’, and suppose r to be the positive num- 
ber denoting the variable distance between them, so that 7 is 
the length of the vector a’ — a, and, therefore, by the princi- 
ples of this calculus, 
riz v¥ {=a —a)}; (2) 
we shall have, by the formula (1), the two equations, 
d*a mr" Va’ mr" 
de. peat. df. alia 
> 
* See Appendix No, IIL., page xxxvii. 
VOL, III, e 
