338 Mr. Youne on an Eatension of a Theorem of Euler, 
ponent terms must be divisible by a’; for no part of one term can be cancelled 
by a part of another, on account of the three independent factors, a, b, c ; and, 
further, on account of these factors, 7,y, and y; must each be divisible by a’; 
therefore, y, itself must be divisible by a; and we see, from the first term, that ° 
x, must also be divisible by a. Hence, dividing each member of the equation by 
a, Mr. Graves’s theorem is the result. 
From the foregoing values for 2,, y,, those for x,, y,, may, of course, be 
readily obtained : but they are already given in Mr. Graves’s paper. 
Belfast, August 2, 1847. 
AppirionaL NoTE REFERRED TO IN PAGE 336. 
I am indebted to the courtesy of Sir William Rowan Hamilton for the following 
communication, respecting the researches of John T. Graves, Esq., and for permission to 
append it to the foregoing Paper. It will be seen that I have had no opportunity of 
making any other use of it. 
Note, by Professor Sir W. R. Hamilton, respecting the Researches of 
John T. Graves, Esq. 
‘© You are aware, from the statement made by me to the Royal Irish Academy, on 
the occasion of my presenting your eight-square formula last summer, and published in 
the Proceedings for that evening (June 14, 1847), that my friend, John Graves, had 
previously sent me an equivalent formula, in a letter dated the 26th of December, 1843, 
which reached me before the end of that year. That letter, indeed, having been written 
in haste, upon a journey, contained a few errors of sign; but these were completely cor- 
rected in a shortly subsequent communication, from which the formula in the Proceedings 
has been transcribed. My present object is to mention that J. 1. Graves, to whom I had 
previously communicated my theory of guaternions, was early led, by his extension of 
Euler’s theorem, to conceive an analogous theory of octaves, involving seven distinct ima- 
ginaries, or square roots of negative unity, namely, four new roots, which he denoted by 
