Sir Witu1am Rowan Hamirton’s Researches respecting Quaternions. 227 
To describe more fully the process which is thus briefly indicated, we may 
observe that, besides the ”* constant coefficients c, there are now given, or sup- 
posed to be known, 2” ordinal relations of the forms a, and a/ (or numbers pro- 
portional to these 27 relations), as the constituents of the two given ordinal sets 
of the " order, q and q’; which sets are here regarded as the divisor set and the 
dividend set respectively. Thus the n’ ordinal relations of the form a,,, are con- 
ceived to be known, as depending in a known manner on the m given relations a,, 
by the m? expressions of the form (135); and on substituting for these 7? ordinal 
relations, and for the 7 other given relations of the form a, in the 2 formule 
(138), any system of numerical values which shall be (in the sense of the 14th 
article ) proportional to these different ordinal relations, we shall thereby obtain x 
linear equations, of an ordinary algebraical kind, between the 2 sought numbers, 
m,: from which these latter numbers may then in general be deduced, by any of 
the usual processes of solution of such ordinary and linear equations. 
For example, after fixing upon any standard ordinal relation, or relation 
between two selected moments of time, and calling it a, we may first prepare the 
equation (138) by putting it under the form, 
a, +> a=Z,.m,(a,, +a); (140) 
in which £, is the characteristic of a summation performed with respect to r, and 
the quotients in both members are numerical. And then, by suitable combina- 
tions of the numerical quotients in the second member of this last equation, which 
combinations are determined by the given expressions (135), we may find a sys- 
tem of n* numerical coefficients of elimination, l,,,, of which the values depend 
on the constant coefficients c, and on the m given numerical quotients of the form 
a,-a, but are independent of the m other quotients a/+ a, and satisfy the n? con- 
ditions included in the formula, 
x, .1,.(a,.+ 2) = 0, or =/, according as r’ 2 or =r; (141) 
/ being here another number, namely, the common denominator of the elimina- 
tion. For in this manner we shall have 7 final expressions of the form, 
Mm; = 1 Z,.1, )(a,-—a)s (142) 
by which the sought coefficients of the symbolical quotient (139) can be, in 
general, determined. 
