244 Sir Witi1am Rowan Hamixton’s Researches respecting Quaternions. 
symbols 7j& in their composition ; provided that we do so without establishing 
any linear relation between those three symbols and unity. This last restriction 
is necessary, in order that each of the four symbolical formule, (226), (230), 
(233), (236), not involving the index s, may be, as we have supposed, equiva- 
lent to the corresponding one of the four arithmetical formule, (224), (229), 
(232), (235), in which that index s, occurs, and is permitted to receive any one 
of the four values, 0, 1, 2, 3. 
25. If we write, for conciseness, 
GW = Nooo == Moy + fone == N39 (238) 
the conditions of the preceding article give the sixteen symbolical equations : 
Yoo =%o3 YIn="Yo3 Yr=IG3 Yuo=hhs 
Qo = Jobs In = "G3 Yo = Jes Ns = Kgs 
av = JoJ 3 Yr =*YoI > Yr =JV0I3 Joa = kod 
Yoo = Tks Yn = Yok > Yr =JJoks Tos = hqoks 
(239) 
in which, while still retaining the linear independence lately assumed to exist 
between 7, j, i, and 1, we may now suppose that the squares and products of the 
three symbols, 7, j, &, are determined, or eliminated, by the help of the funda- 
mental formula (a), assigned in the sixth article, namely, 
C= —e =o — Bs (a) 
together with those others which this may be considered as including, especially 
the following : 
yah f= he eH ey S — 1; AS), k= = 7. (3) 
In this manner, by (225) and (238), while the first of the sixteen symbolical 
equations (239) is identically satisfied, each of the other fifteen will resolve itself 
into four ordinary equations, independent of the three symbols 2, j, & ; and thus, 
if we denote, for conciseness, four of the numerical coefficients of quaternion 
multiplication as follows, 
Nis) SOs Ng NEMO IG Ie, (240) 
the other sixty coefficients of such multiplication may be expressed in terms of 
these: and the values so obtained will satisfy the two hundred and fifty-six con- 
