Sir WitirAM Rowan Hamitton’s Researches respecting Quaternions. 257 
To express the same problem otherwise, with the help of the definition of divi- 
sion, (296), we have now the system of the two equations, 
C=79 Was (298) 
q' and q,, being those two distinct quaternion products which arise from the 
multiplication of the same two quaternion factors, g and q’, with two different 
arrangements of those factors; and we are to eliminate the four constituents of 
one of those two quaternion factors, namely, the constituents of the factor q’, 
between the eight separate and ordinary equations into which the two quaternion 
equations (298) resolve themselves. If we write, for this purpose, 
qa=w+ie + yy + kez, | 
qf aw + iv’ + jy + ke’, | 
f= w+ ba! 4 jy + kz", (299) 
V,= es 0m, Wut kz, 
we shall then have the four equations, 
w’ =uw — ae —yly — 272; | 
a, ant Z—2Y3 
a we + aw +y y | (300) 
yY =wy—az +y'w4+ 72; 
Zswe+ey —yrt+ ew; 
together with the four others which result from these by interchanging, in the 
right hand members, the accented with the unaccented letters, and by changing 
in the left hand members upper to lower accents; namely, the four following : 
w= ww! — a2) yy — 22 ; | 
Rees nee cg (301) 
Y,, = wy — 47 + yw! + 22"; | 
Zz, = we + ay’ —ya' + zu’. 
It thus appears immediately that 
iy = Was (302) 
and the elimination, above directed, of the four numbers w’, x’, y’, 2’, that is, of 
the constituents of the numeral quaternion q’, between the eight equations (300), 
(301), gives these three other equations, which complete the solution of the 
problem, so far as it depends on the above-mentioned elimination : 
2m 2 
