336 Mr. Youne on an Extension of a Theorem of Euler, 
laws: in fact, it is their non-subjection to these laws that alone constitutes their 
novelty. But the author of the present paper makes these remarks on the fun- 
damental character of the quaternion symbols with diffidence, lest he should, in 
any degree, obscure what has been so often and so fully explained by the distin- 
guished propounder of the quaternion theory himself. He will merely observe, 
in conclusion, that a form somewhat more general may be given to the original 
quaternion theorem ; for we may write it thus, viz. : 
(w + the + joy + kbez) (w' + tba’ + jey’ + kbez') = 
w! + iba” + yey” + kbez", 
as is evident from the construction at page 327, /b, /c, (bc), being changed 
into b, c, be. 
Norr.—Mr. John T. Graves’s demonstration of the eight-square theorem, 
and which, I believe, has not as yet been published, was, I learn, conducted by 
aid of the quaternion theory, modified by the introduction of certain additional 
imaginary elements :—I believe, four.* Without such additional symbols it would 
seem that the theorem could not be established in this way; as it is not the case that 
the sum of two quaternions, multiplied bythe sum of two quaternions, produces the 
sum of two quaternions. It would be interesting to see the quaternion calculus 
extended to “octaves ;” and it is to be hoped that Mr. Graves may be prevailed 
upon to make his researches on this subject public. They would probably sug- 
gest a generalization of the coefficients b, c, bc, introduced, in the foregoing 
paper, into the eight-square formula; just as the coefficients here adverted to, 
when employed in the form for fours, might have suggested the quaternions, as 
already noticed above. 
The only more general form for these coefficients which occurs to myself, is 
* See additional note at the end of this Paper. 
