Sir Witu1AmM Rowan Hamitton’s Researches respecting Quaternions. 253 
that they will serve, hereafter, to express the result of a successive multiplication, 
or the continued product of three numeral quaternions. And by applying the 
associative principle, already considered in the twenty-first article, to such suc- 
cessive multiplication, we see that, instead of developing the formula (257) by a 
process which was equivalent to the development of the system of the two 
equations, 
m+ mi+ m+ mk = (a + bi+c7 + dk) (m+ mi+my+m-), (280) 
and 
me mim 7j+ms k= (mommy mk) (m+ m+ mj+ mk), (281) 
we might have developed the same formula (257) by a different, but analogous 
process, founded on a different mode of grouping or associating the three qua- 
ternions which enter as symbolic factors. For we might have introduced this 
other quaternion, 
m,+m>i+tm; 7+ msk= (m+ mit + m7 + mk) (a+bi+ej+dk); (282) 
which would have given the expression, 
mo +m t+ m7 ms k= (my +m i+ mz7j + mzk) (mm +m,7j + mk); (283) 
and then the four values (260), for the four constituents of the final product of 
the three quaternion factors which enter into the second member of the formula 
(257), would have presented themselves as the result of the elimination of the 
four constituents of the intermediate quaternion product (282), between the 
eight following equations : 
ms = ma —m,b — myc —m,d; 
m> =mb + mia +md — mie; 
(284 
m, = me — md +m,a+ mb; ) 
m; = md + m,c — mb + mia; 
my =m, m, — m>m,— M>m,— M; Ms ; 
my = my m, + mm, + m;>ms;— mM;mM, 3 ; 
- (285) 
ms = MyM, — MM; + mz; mM, + Mm; mM, 3 
m; =m, m, + mm, — mm, + m;m,. 
And accordingly, on comparing these eight equations with the four expressions 
(260), we arrive at the same quadrinomial values for the sixteen coefficients 
