250 Sir Witr1am Rowan Hamirton’s Researches respecting Quaternions. 
D,= — mm, + m,m, — mm, — mm, ; 
D, = — mim, — m,m, + mm, — Mm, Mm, 3 if 
1 0 2 3 2 0 3 1? (264) 
Dz, = + mm, — mm, — mm, — MyM, 3 
D,=+ mm, + mm, + m;m, — m;M,. 
And thus may the problem of the multiplication of numeral quaternions be 
resolved, without any restriction being laid on the numerical values of the four 
arbitrary constants, a, b,c, d. The modular equation (251), namely, p’’ = p’p, 
will extend to this more general system, if we define the modulus p of the qua- 
ternion (7p, ™,, m,, m.,) by the formula : 
wa(e+h+ec+ad’) (m? +m? +m, + m,). (265) 
Thus, with the recently established forms (261), ... (264), of the sixteen func- 
tions A,...D,, we must have, as an identity, independent of the values of the 
twelve numbers denoted by the symbols a 6 ¢ dm, m,m, m, mj, m, m; m;, the 
following equation : 
(aA, + 6B, +00, +4D,)’ + (a4, + 5B, +00, +4D,) 
=f (aA, ts bB, + eC, + dD.) oe (aA, “15 bBy+ eC, ac dD.) 
=(@+bh+c+d’) (me +m?e+m?+ mz?) (m?+m?+m,+m,); (266) 
and therefore, independently of the values of the eight numbers m,... 73, we 
must have these ¢ez other equations : 
(me + m? + m? + m3) (m, + m? + m,* + m,’) 
=A, + A+ 4, +4, = By + BY + BS By 
SUy se lise CS alo Sea ea ables 
0=4,2,+ 4,8, + 4,8, 4,8,; 0=4,C, + 4,0, +A, +4.C,; 
0=A,D,+4,D,+A,D,+A,D,; 0=B,C,+B,C,+B,C+B,C,3 (268) 
= B,D,+B,D,+B,D,+B,D,; 0=C,D,+CD,+- CD, + C,Dy 
(267) 
Although these identities admit of being established in a more elementary way, 
yet it has been thought worth while to point out the foregoing method of arriving 
at them, because that method follows easily from the principles of the present 
theory. 
