t 
[3] 
ive) 
Sir Wini1am Rowan Hamitton’s Researches respecting Quaternions. 
wr, + zy, — Y2Z, = ue! 4+ yz" — zy"; 
wy, + z,,— 2a, = wy" + 20" — az"; (303) 
we, ty@,,— ty, = we" + cy” — yx". 
These equations conduct to the relations, 
U2), +YY,, aR ei, = rx" + yy" aE ee, (304) 
and 
Ui by te, aa? tyP +2”; (305) 
which, as it is easy to foresee, will be found to have extensive applications, and 
which may also be easily obtained, by observing that, before the elimination of 
u’, a, y’, 2’, the equations (300), (301) give 
£42" =2(wr + u'r); 2,— 2" = 2(y2 — zy’); 
Y,+y" =2(wy' +u'y)s y, —y" = er’ — #2’); (306) 
B,+-2 = Wwe +w'z); 2, — 2" = 2(ay’ — yz’). 
33. Although these latter combinations (306), of those equations (300), (301), 
conduct without difficulty to the equations (303), (304), (305), yet it is still more 
easy, when once the principles of the present theory have been distinctly com- 
prehended, to deduce the last-mentioned equations, by treating in the following 
way the problem of the foregoing article. 
Instead of resolving the numeral quaternion q/ into the four separate terms, 
w’, ix’, jy’, kz’, as is done in the second of the four expressions (299), and then 
eliminating the four constituent numbers w’, 2’, y’, 2’ between the eight ordinary 
equations into which the two quaternion equations (298) resolve themselves, we 
may eliminate the quaternion q/ itself between those two equations (298), and 
so obtain immediately, without any labour of calculation, this new quaternion 
equation, 
WT = 94 (307) 
which, by the three remaining expressions (299), and by the equality (302), 
becomes : 
(tz, + jy,, +hz,,) (wir +jy + hz) = 
(we + jy + kz) (ix + yy" + kz’). (308 ) 
If now we perform the multiplications here indicated, attending to the funda- 
mental expressions (a) (B), for the squares and products of the three symbols, 
