290 Sir Wiiu1am Rowan Hamitton’s Researches respecting Quaternions. 
But also, by the general analogy of the present notation, if we denote by j/’ and 0// the 
modulus and amplitude of the same reduced product (31), we shall have 
_— pe” cos 9/1 = w" =p’ cos 0”, sin 8/=r/"; (36) 
therefore, 
/’= pp! ¥ (cos 6? + cos 0 + cos ’?— 2 cos f cos 8 cos 8) ; (37) 
and 
yr 
cos 0/’= coal (38) 
¥ (cos 6*+ cos @” + cos 6” —2 cos 6 cos 6! cos 0”) 
Again, by (17), (28), (34), (36), (37), 
r/'= V (Ww? =p!) = pw’ v (1 +2 cos 8 cos 6 cos 6” — cos 6? — cos 8 — cos ion 5 (39) 
= py’ sin @ sin 0 sin 77’ ; 
an expression for the radius of the pure imaginary triplet, 
70, f! +JY)( 4 RZ,/'5 (40) 
that is, of the complete product (4) minus the reduced product (31), which agrees with 
the second equation (30), because, by spherical trigonometry, 
sin @sin @ sin77’=sin 6” sin r’r/’ ; (41) 
and which gives 
p= pu! V (A—(sin @ sin & sin 77’)’). (42) 
We might call the triplet (40), (which remains when we subtract the reduced product 
from the complete product), the residual triplet, or simply, the residual, of the product of 
the two proposed quaternions (4). And we see that this residual is always perpendicular 
to the reduced product, when it exists at all; for we shall find that it may sometimes 
vanish. It is the part of the complete product which changes sign when the order of the 
factors is changed. 
‘* These remarks on the geometrical construction of the eguations of multiplication (5) 
and (6) have, perhaps, been tedious ; they certainly are nothing more than deductions from 
those equations, and, consequently, from the fundamental assumptions (1), (2), (3). Yet 
it may not be altogether useless, in the way of illustration, to draw some corollaries from 
them, by the consideration of particular cases. 
‘* Multiplication of two Reals.—It is evident from the figure that, as [the two internal 
angles] @ and 6’ tend to 0, [the external angle] 0” tends to 0 likewise; and that the same 
thing happens with respect to 0”, when @ and @/ both tend to 7. Hence the product of 
two positive or two negative real quantities is a real positive quantity. But when one of 
the two amplitudes of the factors, 0 or 6’, tends to 0, and the other to z, then 6” also 
tends to 7; the product of two reals is, therefore, real and negative, if one of the two 
factors is positive and the other negative. 
“* Multiplication by a Real.—If 0 tend to 0, & tends to become = 6’, and r” tends to 
