224 Sir WitrtaAm Rowan Haminton’s Researches respecting Quaternions. 
and positive or negative, or null, from — © to + o, without violating any of 
the usual rules for operating on such numbers, by addition, subtraction, multipli- 
cation, and division; or rather we might deduce anew all those known rules for 
those fundamental operations on what are usually called real numbers, as conse- 
quences of the foregoing formule, or as necessary conditions for their generaliza- 
tion; observing, indeed, that for the case of ¢xcommensurable (but still real) 
multipliers, whether operating on a simple ordinal relation a, or on an ordinal 
set q, we are to use also an equation of limits, of the form, 
(lim. m) X a= lim.(m X a). (126) 
It is a consequence of these conceptions and notations that an ordinal set q is 
multiplied by a number m, when each of its constituent ordinal relations, k,4q, 
nq, &e., is separately multiplied thereby ; so that we may establish the formula, 
m/(a, b, ¢, . -) = (ma, mb, me, . .); (127) 
and therefore also, 
Ry.mq = MR,qQ3_R,.mq = mR,q ; &e. (128) 
And any ordinal relations, such as ma, mb, &c., or any ordinal sets, such as mq, 
mq’, &c., which are thus obtained from others, such as a, b, &c., or q, q’, &c., by 
multiplying them respectively by any common number m, may be said to be 
proportional to those others. 
We may also say that any ordinal relations, such as ma, ma, &c., and that 
any ordinal sets, such as mq, mq, &c., are proportional to the multiplying num- 
bers m, m’, &c., by which they are generated from any common relation a, or 
set q, as from a common multiplicand, when such generation is possible. 
Case of Existence of a simple numeral Quotient, obtained by a particular Divi- 
sion of one ordinal Set by another. 
15. The recent theory of the mz/tiplication of an ordinal set by a number, 
enables us to assign, in one extensive case, an expression for the result of the 
division of one ordinal set by another; for if we regard the equations 
(a’+a)Xa=a’, (q+ q)Xq=q; (129) 
as being identically or definitionally true by the general symbolical correlation of 
