240 Sir Witt1am Rowan Haminton’s Researches respecting Quaternions. 
of derivation and of multiplication, admits of being interpreted or accounted for 
in a very simple manner. 
The coefficient c,,,.,, introduced in the sixteenth article, may be regarded as 
having been generated, or, at least, brought under our view as follows. We first 
supposed an ordinal set, q, to be operated on by the elementary characteristic of 
derivation X,, so as to produce thereby a derivative set, q,. We then operated 
on this derived set, in a way which may be indicated by the characteristic of 
ordinal separation, k,, and so obtained a result of the form 
Rs Xrq = a, (217) 
And, lastly, we analyzed this result, so as to find the part of it which depended 
on, and arose from, the constituent a, or r,q of the original operand set ; and 
the coefficient of this constituent a,, in the part obtained by this analysis, was 
denoted by c,,, , and was regarded as a coefficient of derivation. On the other 
hand, the coefficient of multiplication, 7, may be said to arise thus: an ele- 
mentary derivation, denoted by X,, is succeeded by another, denoted by x, 3 
the compound operation, X,X,, is detached from the operand, and regarded as 
equivalent to a single complex derivation, of which the characteristic may be 
symbolically equated to a certain numeral set; this last set is subjected to the 
characteristic of numeral separation N,, or to an analysis equivalent thereto; and 
the result is, by (212), the coefficient of multiplication in question. 
Now the agreement of the results of these two processes, which is expressed by 
the equation (215), becomes quite intelligible and natural, if we conceive that 
the constituent a, of the operand set q, on which constituent alone we really ope- 
rate in the former process, the others being, in fact, set aside, as contributing 
nothing to the result here sought for, has been itse/f produced or generated by 
an earlier operation of the form a, X, (where a, has the same signification as in 
(184)), from some one primary or original ordinal relation, such as that which 
was denoted in some recent articles by the letter a. In this manner we may be 
led to look upon any ordinal set, such as the set q in the equation (133), as 
being generated by a certain complex derivation, which is expressed by a certain 
numeral set g, from a single standard ordinal relation, a, or from the relation 
between some two standard or selected moments of time, according to either of 
the two reciprocal formule : 
q=7a=X.a Xa; Or, J=QtaH—»,.a Xr; (218) 
