181 
Monpay, Fesruary 26TH, 1855. 
THOMAS ROMNEY ROBINSON, D.D., Presipen’, 
. in the Chair. 
Rev. Proressor 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,—Y ou 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), 
S ajnn” = (1, m, n) vk” 5 (1) 
where /, 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 
(l+m+n)!_ 
Ni, =, 9= Tata 
(2) 
q each of these N arrangements gives (by the rules of 7h) a pro- 
duct =+1. aymk"; and the sum of all these positive or nega- 
tive unit-coefficients, + 1, thus obtained, is the numerical co- 
efficient denoted by (/, m, 2). 
___ * Since each arrangement must have 7 or 7 ork to the left, 
we may write, 
Sym he = tS MR” 4 PSTN + hVUymh (3) 
_ and it is easy to see that the coefficient (/, m, m), or the sum 
_ = 1), vanishes, if more than one of the exponents, J, m, n, be 
_ odd. Assume, therefore, as a new notation, 
7 (2A, 2u, 2v) =(A, pw v}3 (4) 
=) VOL. Vi. R . 
