Sir Wiri1am Rowan Hamitton’s Researches respecting Quaternions. 249 
This formula resolves itself, by those relations, and by the linear independence of 
7, J, k, and 1, into four separate equations, which may be obtained from the four 
equations (248), by changing m,, ,, m,, ™,, respectively, to 
my, = am, — bm, — em, — dm,; 
m, = am, + bm, + em, — dm,; (258) 
m; = am, — bm, + em, + dm,; 
m3; = am, + bm, — em, + dn,; 
so that, with these abridgments, the four equations included in the formula (257) 
may be thus written : 
mM, =m,mM, — M,m; — Mm, mM; — m,m3; 7 
mi = mom, + mm, + mm; — mim3; | (259) 
m,) = mm, — mim; + mm), + mi, m} ; % 
ms; =m, mM; + mm; — mm; + mim). 
In this manner we should obtain the four expressions : 
m, =aA,+b6B,+cC,+dD,; 
m =aA,+bB,+cC,+dD,; 
m;, = aA,+ 6B,+cC,+dD,; (200) 
sie m; =aA,+6B,+cC,+dD,; 
A, = mm, — mm, — mm, — M;M;;3 
A, =m m, + mm, + mm, — m,m, 3 | (261) 
A, = mm, — mim, + mm, + m,m,; 
A,=mm,-+ mm, — mm, + m,m,3 
B,= — mm, — mm, + mm, —'ms m,; 
B, = + mm, — mm, + mim, + mim, ; (262) 
B, = — mm, — mim, — m,m, + mm; | 
B,= + mm, — m,m, — m,m, — m,m,; 
C= — mm, — mm, — mm, + mim, ; 
C,= + mm, — mm, — mm, — mim, ; : 
Cy = mbm, bmi m, — mim, + mi mg ee) 
C,= — mm, + mim, — mm, — mym,; 
223.2 
