334 Mr. Youn on an Extension of a Theorem of Euler, 
a formula which has been otherwise established by Cauchy, in his “ Cours 
Diy 
d’ Analyse,” page 455. Cauchy adds, that when —, —, &c., are equal, the second 
member is reduced to its first term. But this is only saying that f(mr)?= 
J( mx). The foregoing equation warrants the inference (7), as already noticed. 
3. Before concluding these researches, the author is desirous of adding a 
word or two on Sir William R. Hamilton’s Quaternions; a subject which, as 
already remarked at the outset, is intimately connected with some of the specu- 
lations in the present paper. 
If we refer to the four rows of combinations to which we have been conducted 
by the forms (w'+/b.a°+ Yc.y/+V(be).2) (w+Vb.c+Ve.y+ Vv (be).z), 
at page 327, we shall perceive that, with the exception of the signs, these combi- 
nations make up the actual product which would arise from multiplying those 
forms together as factors. This circumstance is calculated to suggest the inquiry, 
whether, by imposing certain laws of combination, in reference to the coefficients 
Vb, Yc, Y(be), the rows alluded to might not be made to represent the pro- 
duct, signs and all. Such an inquiry would lead us to the conditions originally 
proposed by Sir William Hamilton; for, changing /b, Yc, (bc), into 7, 7, k, 
we should find that the laws of combination to which these symbols must be sub- 
ject, to produce the desired effect, are those implied in the following relations, 
Viz. : 
?=pP=kh=-1; 
alas Je =o? l= 9)e 
fi=—k; kj=-1; tk=—- 7. 
With these symbols, under these relations, the factors referred to are quater- 
nions ; and we see that the product of two quaternions must produce a quater- 
nion. And this is the fundamental theorem from which Sir William Hamilton 
has deduced so many and such remarkable results. 
If we were to content ourselves with a very narrow and imperfect view of 
the office of the above symbols, we might regard them simply as ingenious con- 
trivances for facilitating the construction of the four-square formula; since, after 
the combinations which make up the product of two quaternions have been 
obtained, conformably to the foregoing relations, we may then change our tem- 
porary symbols back again to their originals, /b, Vc, ¥(bc), and thus easily 
