Sir Witt1am Rowan Hamirton’s Researches respecting Quaternions. 225 
the signs X and ~, we may then write, in virtue of the formula (127), this other 
and correlative formula, 
(as bisc5. -)' "(ay Dy Gee) 7, (130) 
whenever the following conditions are satisfied : 
GY A Sy oy) SS SU, (131) 
In other words, we know how to énterpret the quotient q' ~ q, of one ordinal set 
q’ divided by another q, namely, as being another expression for a simple or 
single number m, in the case when the constituent ordinal relations of the one 
set are proportional (in the sense lately defined) to then homologous constituents 
of the other set; and we have, 7m that case, the continued equation, 
q +q=R,q/ + Rg = RQ’ + RQ = &e. (132) 
But in the infinitely many other cases in which this condition of proportionality 
is not satisfied, the m numerical quotients, r,q’ + R,q; R,q’ + Rg, &c., being at 
least partially different among themselves, and therefore being not each equal to 
one common number m (whether commensurable or incommensurable, and 
whether positive or negative or null), it is, for the same reason, impossible to find 
any ONE such number, m, which shall be correctly equated to the quotient q’ + q 
of the two proposed ordinal sets, in consistency with the foregoing principles. It 
is, however, not impossible to find a system of numbers, which may, consistently 
with those principles, be regarded as representing this quotient of the division of 
one ordinal set by another ; and we proceed to give an outline of a process by 
which such a numeral system, or complex quotient, may be found. 
Investigation of a complex numeral Quotient, resulting from the general 
symbolical Division of one ordinal Set by another. 
16. Conceive that from any proposed expression of the form, 
“= (pete ctinoe ts,)) (133) 
for an ordinal set q of the n™ order, we form m other expressions of coordinate 
derivative sets, qo. Gy +» Qn» according to the type, 
S< 5 qh S<Fiqi=q7. = (Garp ae emia tan ae pe) (134) 
2H 2 
