242 Sir Wittram Rowan Hamivton’s Kesearches respecting Quaternions. 
We might content ourselves with proving the truth of this assertion by actual 
arithmetical substitution of these sixty-four values in the two hundred and fifty- 
six equations; but the following method, if less elementary, will probably be 
considered to be more elegant, or less tedious. It will have, also, the advantage 
of conducting to a somewhat more general system of expressions, by which the 
same equations can be satisfied; and will serve to exemplify the application of 
the fundamental relations, (a), (B), which were assigned in the sixth and eighth 
articles, between the important symbols 774, and on which the present Theory of 
Quaternions may be regarded as essentially depending. 
24. Let us, then, first form, from the type (219), by changing the index r to 
the value 0, the following less general type, which, however, contains under it 
sixty-four out of the two hundred and fifty-six equations of condition to be 
satisfied : 
Nowa Mtoe + Nour Mes 1 Nous Mize 1 Nous Meas 
= Nous M100 + Mus Mm F- Nous Moz  MRsuz M03 (224) 
Make, tor abridgment, 
Yiu = Nw + Ma + nus + knys 5 (225) 
ijk being the three symbols just now referred to; we may then substitute for 
(224) the following formula, deduced from it, but not involving the index s: 
Now Jo + Nona Yr $ Now Yeo + Nous Us 
= Jou M100 Jiu Mr F You Mos + Yu Ns: (226) 
This, again, will reduce itself, by the same definition (225) of the symbol ¢,,, to 
the identity, 
You {to = You Year (227) 
and therefore will be satisfied, if we satisfy the six conditions : 
Wa = Wo 3 2 =99a3 da = kqu3 | (228) 
Viu = Youd 3 Tou = Youd 3 Y3u = Youk- J 
If, instead of making 7 = 0, we make 7 = 1, in (219), we then obtain, instead of 
(224), the formula : 
Nywo Mos A Miu Mas FF Miuz Mes + Mis Nese 
= Nous Mo A Mius Mer Maus Mere A Maus Me 33 (229) 
