Sir Wirt1am Rowan Haminton’s Researches respecting Quaternions. 243 
and the symbolic equation (226) is replaced by the following : 
Nywo Jeo 7 Mur Ya 7 Mie Yeo + Mins Yes 
= Youn + Qiu Mar + QauM a2 + Yau Muss (230) 
which, under the conditions (228), becomes first, by the definition (225), 
Yiu Yoo = Jou Yas (231) 
and then is seen to be satisfied, in virtue of the same conditions. 
In like manner by making 7 = 2, in (219), we find 
Nou Nos Nour Mas 7 Nous Nis + Nous Nias 
= Nous M20 = Mins Me. FH Nous Moe $+ Ngus N13 3 (232) 
and this, under the form 
Nou Yt + Nou Ya + Nova Te + Noys Yes 
= Jou M20 + Qiu Mer + JouT22 + YauMr239 (233 ) 
is satisfied by the same conditions (228), since they give 
Jou Gto = You Yt2 (234) 
Finally, the formula obtained from (219) by making r = 3, namely, 
M3uo Nios A Maur Ns FF Maua Nias +H Ngug Miss 
= Nous N30 $F Mirus M31 Nous Nise TF gus Maa» (235) 
or this other, deduced from it by the help of (225), 
N3u0 Yto 7 Maa Yr + R3u2 Yor + N3us Yes 
= Fou + Jiu t31 + Jou Nese + T3u 11339 (236) 
is satisfied by the same conditions (228), which give 
Yau Yoo = You Yes- (237) 
We shall therefore satisfy not only the sixty-four arithmetical conditions included 
in the type (224), but also the sixty-four others included in the type (229), the 
sixty-four included in (232), and the sixty-four included in (235); that is to 
say, we shall satify the whole system of the two hundred and fifty-six arithme- 
tical (or ordinary algebraical) conditions included in the formula (219), if we 
satisfy the system of the siz symbolical equations (228), which involve the three 
