= 
Sir Witi1am Rowan Hamirton’s Researches respecting Quaternions. 247 
left hand factor in the expression for a product) is the quotient obtained by 
dividing the product by the multiplicand; and may write the formula, 
(mi/, mi’, ms’, mz ) 
/ / / / 
Mo, Mi, M3, 3) = : 
Os ) (My Ms M,, Ms) 
(249) 
It is easy to see that if we make, for abridgment, 
wom, +m? +m, +m, | 
we = me +m? + me? + m2, (250) 
= my? m?+ msy?+ my”, | 
and regard p, py’, . as positive (or absolute) numbers, the equations (248) give 
the following very simple but important relation : 
pikes Tir (251) 
If then we give the name of modulus to the (positive or absolute) square-root 
of the sum of the squares of the four (positive or negative or null) numbers, 
which enter as constituents into the expression of a numeral quaternion, we see 
that it is allowed to say, for such quaternions (as well as for couples and their 
analogous moduli), that the modulus of the product is equal to the product of 
the moduli. The equations (248) give also, for the numerical constituents of 
the quotient (249), the expressions : 
mo = p?( mim, + mim, + msm, + m;'m,) ; 
m, = pw ?(—mi'm, + mim, — ann + m;'m,) ; | (252) 
m; = p*(—m4'm, + m;'m, + m,’m, — m3'm,) 3 ny 
ms = w*(—m,'m, — mm, + msm, + m;m,) ; | 
which may be compared with the expression (183) for the quotient that results 
from the division of one couple by another. As a verification, we may observe 
that they give, as it is not difficult to see that they ought to do, 
aid My, My ii) iGO Oa: (253) 
And these results respecting products and quotients of two numeral quaternions 
may easily be remembered, or reproduced, if we observe that we have the fol- 
lowing general expression for a numeral quaternion : 
g=(m, mM, mM, m,) = mM, + mm, + 7m, + km,; (254) = (c) 
VOL. XXI. 2. 
