181 
Monpay, Fresruary 26TH, 1855. 
THOMAS ROMNEY ROBINSON, D.D., Presipenv, 
in the Chair. 
Rev. Prorsessor Graves communicated the following extract 
from a letter addressed to him (under date of January 26th, 
1855) by Sir William R. Hamilton :— 
«* My pear Graves,— You may like, perhaps, to see a way 
in which I have to-day, for my own satisfaction, confirmed (not 
that they required confirmation) some of the results announced 
by you to the Academy on Monday evening last. 
«¢ Let us then consider the function (suggested by you), 
= ama" = (1, m, n) vk ; (1) 
where J, m, n are positive and integer exponents (0 included) ; 
the summation & refers to all the possible arrangements of the 
1+m+n factors, whereof the number is 
oe (2) 
each of these N arrangements gives (by the rules of 7k) a pro- 
duct =+1.7j"k"; and the sum of all these positive or nega- 
tive unit-coefficients, + 1, thus obtained, is the numerical co- 
efficient denoted by (/, m, n). 
«< Since each arrangement must have 7 or 7 ork to the left, 
we may write, 
> amRn Be ida ymhn + Samper aL heap mes c (3) 
and it is easy to see that the coefficient (2, m, n), or the sum 
= (+ 1), vanishes, if more than one of the exponents, J, m, n, be 
odd. Assume, therefore, as a new notation, 
(2A, 2u, 2v)={X; pe v}s (4) 
VOL. VI. R 
