246 Sir Wit1am Rowan Hamutton’s Researches respecting Quaternions. 
for the results of operating, by the four elementary characteristics of derivation, 
Xo. Xiy Xo) Xz which are thus seen to be equivalent to 1, 2,7, &, on the ordinal 
° 
quater nion, 
C= (aps > Ags a,). (55) 
Whatever the constituents of this original operand may be, since the equations 
of detachment have been satisfied by the choice of the constant coefficients, we 
shall have, by the formula (153), and by the values (220) .. (223), sixteen 
expressions for the symbolic squares and products of these elementary charac- 
teristics of derivation, which are independent of the quaternion first operated on ; 
namely, the sixteen expressions following : 
5 ig Sig ral OF pa eer ae ooh ae Es Es See 
Kg, = ae Xa ns Oe Kage Oa xX pS ee (245) 
Xo Xe Mase hee he RES ee Xe ee 
Xo X3= X%g3 Xi X3=—%os Xa Xs Xp; SE pe “UNE 
which might also be deduced from the equations, 
SOE Sa aX ies (246) 
Product and Quotient of two numeral Quaternions ; Law of the Modulus. 
27. We may also write, by (155), 
(my, m;’,m;, m;,.) = (mi, Mig My M5) (Myy My May 1205 )s (247) 
and may say that the nwmeral quaternion (mj, mj’, m;', m3) is equal to the 
product obtained when the numeral quaternion (m,, m,, m., m,) is multiplied, as 
a multiplicand, by the numeral quaternion (7, 7}, m, m3) as a multiplier ; pro- 
vided that, by the formula (149), with the same values of the coefficients of mul- 
tiplication, we establish the four following equations between the twelve numerical 
constituents of these three numeral quaternions : 
mo = mm, — mM, mM, — MM, — M3 M, 5 
mm; = m,m, + mm, + mm, — m3 M, 3 
mi, = mm, — mm, + mm, + m3 mM, 3 
m, = mm, + mm, — m,m, + m;m,. 
(248) 
Under the same conditions we may say that the matiplier quaternion (or the 
