Sir Wirt1Am Rowan Hamirton’s Researches respecting Quaternions. 237 
may be independent of the original ratio of a, to a,, or of a, to a,; and each of 
the twenty-four equations thus furnished, of which the equations (201), (202), 
(203), are three, is an equation of the third dimension, with respect to the coef- 
ficients of derivation and multiplication, Cj.) &C., Mo, &e. | We should, therefore, 
by this method, have obtained equations more numerous and less simple than 
those which were given by the method of the eighteenth article: which method 
there is, therefore, an advantage in introducing, even for the case of couples, 
and much more for the case of quaternions, or other ordinal and numeral sets ; 
although the method above exemplified appears to offer itself more immediately 
from the principles of the seventeenth article. 
But to exhibit by an example the agreement of the two methods in their 
results, let the symbols defined by the equations (163), (164), be employed to 
abridge the expression of the equations (201), (202), (203); the latter will then 
become : 
e (ad — ch) =d(a’+a’/b) —ch(a+b'); ] 
e(ad' — ch’ +a'd — cb) =d' (a + a'/b) — cb (a+d’) | (204) 
+ da’ (a +0) — c(ba' +8”); 
e(a'd'— cl‘) = da (a+b’) —c' (ba +b”); J 
and it is evident, upon inspection, that these three equations (204) may be 
deduced by elimination of e’ from the four equations of detachment (165), which 
were obtained by the simplified method; and which, in that method, formed part 
of a system of only sixteen (instead of twenty-four) equations, each rising no 
higher than the second (instead of the third) dimension. 
Associative Principle of the Multiplication of numeral Sets: Characteristics of 
numeral Separation. 
21. Whenever, for any value of the exponent m of the order of a set, we 
have succeeded in satisfying the z* simplified equations of detachment, included 
in the formula (160) of the eighteenth article, and have thereby found a system 
of n’ coefficients of derivation, and a connected system of 7’ coefficients of mul- 
tiplication, with reference to which two systems of coefficients an equation, or 
rather a system of equations, of the form (153) can be established, independently 
of the x — 1 ratios of the constituents of that ordinal set q, on which the two 
