276 
is found to be satisfied, we may then infer that the five cor- 
ners A, B, C, D, E, of this pentagon, are situated on the surface 
of one common sphere. This equation of homosphericism 
(5) or (6), appears to the present author to be very fertile 
in its consequences. To leave no doubt respecting its meaning, 
and to present it under a form under which it may be easily 
understood by those who have not yet made themselves mas- 
ters of the whole of the theory, it may be stated thus: if we 
write for abridgment, 
ay = t (2 — &) +) (Yi — Yo) +h (ZH — %)s 
dy = 1 (Xy — 03) +) (Y2 — Ys) + h (22 — 2), 
a; = 1(X3 — %) +) (Ys — Ys) + (Zs — 24)5 (7) 
a, = 1 (&s— %5) +) (Ys — Ys) +h (Zs — 25)s 
a, =t(t@ —%) +7 Ys—Mm) + k(Z5 — A); 
and then develope the continued product of these five expres- 
sions, using the distributive, but not (so far as relates to ijk) 
the commutative property of multiplication, and reducing the 
result to the form of a quaternion, 
@, Gg G3 dg a5 — w a 1x + yy — kz, (8) 
by the fundamental symbolical relations between the three 
coordinate characteristics ijk, which were communicated to 
the Academy by Sir William Hamilton in November, 1843, 
and which may be thus concisely stated : 
PSP mye — Te (A)* 
and if we find, as the result of this calculation, that the term 
* These fundamental equations between the author’s symbols i, j, 4, appeared, 
under a slightly more developed form, inthe number of the London, Edinburgh, 
and Dublin Philosophical Magazine for July, 1844; in which Magazine the 
author has continued to publish, from time to time, some articles of a Paper on 
Quaternions; reserving, however, for the Transactions of the Royal Irish 
Academy, a more complete and systematic account of his researches on this 
extensive subject. 
