Sir Witt1am Rowan Hamixton’s Researches respecting Quaternions. 229 
for all the * quotients. The value of each of these numerical quotients (150) 
will, in general, depend on the z —1 ratios of the constituents a,, a, ...a,_, of 
the first proposed ordinal set q, or the ratios of the numbers to which these x 
ordinal constituents are proportional ; but it may be possible to assign (at the 
outset) such values to the constant but arbitrary coefficients of derivation c, or 
to subject those m* coefficients to such restrictions, that these n —1 arbitrary 
ratios of the n constituents a, in the expression (133), shall have no influence on 
the value of any one of the n® numbers included in the expression (150). When 
this last condition, or system of conditions, is satisfied, we are allowed to detach 
the characteristics of the successive symbolical multiplications of an ordinal set 
rom the symbol of the original multiplicand ; and as the result of the comparison 
of the formule (136) and (143), and of (147) under the form, 
q’ = (my KX +. + M1 Xn-1) | (151) 
we may write, 
Mg’ Kot «EM Kn—1= (MX oH + Mn Kur) (MyXaHs Mn 1Xn-1)3 (152) 
which will denote the reduction of a system of two successive and complex deri- 
vations, or symbolic multiplications of the kind (136), to one complex derivation 
of the same kind. Under the same conditions, the successive performance of two 
simple or elementary derivations, of the kind (134), will be equivalent to the 
performance of one complex derivation, of the kind (136), with numerical co- 
efficients independent of the original derivand, as follows : 
XK Kr S Vy Myr,r” Kom (153) 
We may also regard the m variable numerical coefficients m,, in the quotient 
(139), obtamed by the symbolical division of one ordinal set by another, as com- 
posing, under the same conditions, a NUMERAL SET; and this new sort of se¢ may 
be detached, in thought and in expression, from the two ordinal sets which have 
served, by their mutual comparison, to suggest it. The quotient (139), when 
thus regarded as a numeral set, may be denoted as follows : 
q =q=97 =(m, mM, ..™,_,)3 (154) 
the letter g, when used as a symbol of such a set, being written in the Italic cha- 
racter: and then the 2 numerical relations, which are included in the formula 
(149), may be supposed to be otherwise summed up in the one equation : 
(Mo's. Mig. May) = (MG. My MG)! (Mb) * «My v's My _,): (155) 
