126 
ON CERTAIN RESULTS RELATING TO QUATERNIONS. 
[20 
On the contrary, the result of this elimination is the very different equation 
m~*.0 — m'~ l . (f)' = 0 (15), 
equivalent of course to four independent equations, one of which may evidently be 
replaced by 
Mm. M<p' — Mm'. M<p = 0 (16), 
if Mm, &c. denotes the modulus of m, &c. An equation analogous to this last will 
undoubtedly hold for any number of equations, but it is difficult to say what is the 
equation analogous to the one immediately preceding this, in the case of a greater 
number of equations, or rather, it is difficult to give the result in a symmetrical form 
independent of extraneous factors. 
I may just, in conclusion, mention what appears to me a possible application of 
Sir William Hamilton’s interesting discovery. In the same way that the circular 
functions depend on infinite products, such as 
xU(l + —V &c, (17), 
\ mirj 
{:m any integer from oo to — x , omitting m = 0} 
and the inverse elliptic functions on the doubly infinite products 
#n ( 1 + 
. , &c. 
mw + nmi/ 
[m and n integers from x to — x, omitting m = 0, n = 0}, 
(18) 
may not the inverse ultra-elliptic functions of the next order of complexity depend ou 
the quadruply infinite products 
xl\ 1 + 
mw + nmi + o<f>j +p^Jrk 
{m, n, o, p integers from x to — x, omitting m = 0, n = 0, o= 0, p = 0}. 
•(19) 
It seems as if some supposition of this kind would remove a difficulty started by 
Jacobi (Crelle, t. ix.) with respect to the multiple periodicity of these functions. Of 
course this must remain a mere suggestion until the theory of quaternions is very much 
more developed than it is at present; in particular the theory of quaternion exponentials 
would have to be developed, for even in a product, such as (18), there is a certain 
singular exponential factor running through the theory, as appears from some formulae 
in Jacobi’s Fund. Nova (relative to his functions ®, H), the occurrence of which may 
be accounted for, d priori, as I have succeeded in doing in a paper to be published 
shortly in the Cambridge Mathematical Journal [24].