Sir Witt1am Rowan Hamitton’s Researches respecting Quaternions. 287 
factors, while the two adjacent angles are the amplitudes of those factors, the remaining 
angle will be the supplement of the amplitude of the product. 
«© Combining (5) with (6), we find that 
ww! + ax" +yy" 22 =(w*+ xr +y¥ +2°)w'; (22) 
we! + 0a! yy" 4 2/2" = (w he? + y? +2”7)w ; 
therefore, by (16) and (17), 
cos 0’ = cos @” cos 0 at 0” a 6 {cos p” cos p + ae p” sing cos(h —”)}; (23) 
cos 8 =cos 6” cos 6’ + sin 8” sin 6’{ cos p” cos ¢’+sin p’ sin 9! cos(y’—W) }; 
so that in the spherical triangle lately mentioned, the two remaining sides are the inclina- 
tions of the two factors to their product. This spherical triangle may, therefore, be con- 
structed by merely joining the points r, rn’, rR”, where the sphere, with radius unity, and 
with centre at the origin of 2, y, z, is met by the directions of the radii, r, 7’, 7’, of 
the two factors and the product. The spherical coordinates of these three points are 
0; o> W; ¢”, 5 the spherical angles at the same points are 0, 6’, 7—@’. In the 
solid corner, at the origin, made by the three radii r, 7’, 7’, whatever the lengths of these 
radii may be, the three dihedral angles are 
rrr =03 rrr a0 3 rr'ra7—0'; (24) 
that is, they are the amplitudes of the factors, and the supplement of the amplitude of 
the product. 
‘© Though this theorem of the spherical triangle, r, r’, Rr”, or solid corner, 7, 7’, 7”, 
when combined with the law of the moduli (u/’=p'), reproduces four relations between 
the four constituents, w’, a”, y’, 2’, of the quaternion product, and the eight constituents 
of the two quaternion factors, namely, w, x, y, z, and w’, x’, y’, 2’, that is to say, the two 
relations (5) and (15), and the two relations (22); yet it leaves still something undeter- 
mined, with respect to the direction of the product, which requires to be more closely con- 
sidered. In fact, we can thus fix not only the modulus, ,”, and the amplitude, 9”, of the 
product, but also the inclinations of its radius, 7”, to the two radii, 7 and 7’; but the con- 
struction, so far, fails to determine on which side of the plane rr’ of the radii of the factors 
does the radius of the product lie. In other words, when we deduced the relations (15) 
and (22), we may be considered as having employed rather the equations (9) and (13), 
which were derived from (8), than the equations (8) themselves; the three quantities, 
2,/', y,/, Z,/, might, therefore, all change signs together, without affecting the law of the 
moduli, or the theorem of the spherical triangle. And the additional condition, which is 
to decide between the one and the other set of signs of these three quantities, or between 
the one and the other set of signs in the expressions 
polls ior et oe lis napa ap OEE ay |i eee Hirt ll (25) 
VOL. XXI. 200 
