226 Sir Witt1am Rowan Haminton’s Researches respecting Quaternions. 
in which it is supposed that the constituent ordinal relation a,,,, of the derivative 
set q,, has a determinate and known dependence on the 7 constituents, such as 
ay, of the proposed set q; and let us conceive that this dependence is expressed 
by a formula such as the following : 
Deg Cn Oop ete Cn eran soe tO men anenTs (135) 
the n* coefficients of coordinate derivation, c,. .,., bemg all regarded as constant 
and known numbers, whether positive or negative or null. It will then be pos- 
sible, without altering the constant numerical values thus supposed to belong to 
these n° coefficients, ¢,., to form a complex and variable derivative q’ of the 
set q, by multiplying each of the 7 simple or elementary derivatives already ob- 
tained, such as q,, by a variable number m,, and adding the » products together ; 
and the resulting set may be denoted thus : 
(ay Xo +m, X,+-- + Mm, Meapoc + mM,_; Xn) q I (136) 
> Mo + 7,4) Bp cata Mrqr + +. + Mn_, Qn-1 = q’ > } 
where we shall have 
, 
if (Hip tip oe ge ek) (137) 
if we make, for abridgment, 
a, = My ay, e 2, 8,,2 +--+ Mp ays + -- $+ Mn, Any, 53 (138) 
and the entire collection of signs of operation, m, X, + &c., which is prefixed 
between parentheses to the symbol q in the first line of the formula (136), may 
be said to be a characteristic of complex derivation, or a complex symbolic multi- 
plier. But instead of thus conceiving the set q’ to be deduced from q by this mode 
of complex derivation, or symbolical multiplication (136), with the assistance of 
the constant coefficients of derivation c, and of 2 given values for the variable 
multiplying numbers m, we may inquire, conversely, what system of numerical 
multipliers, ms. . My, +. Mn_,, must be assumed, in order to produce or generate a 
given ordinal set q/, as the symbolical product of this sort of multiplication; the 
multiplicand set q, and the constant coefficients c, being still supposed to be given. 
This inverse or reciprocal process may be called the symbolical division of 
one ordinal set by another, namely, of the set q’ by the set q; and it may be 
denoted by the following formula, which is the reciprocal or inverse of the for- 
mula (136): 
q’ = q = mM, Xo mM, Xe. May Kar (139) 
