230 Sir Wint1am Rowan Hamixton’s Researches respecting Quaternions. 
And conversely, this last equation, which asserts that the numeral set in its first 
member is equal to the symbolical product of the two numeral sets in its second 
member, may be considered to receive its ¢néerpretation from the formula (149) ; 
in which the 2? numbers 7,,,.,,, may be called the coefficients of multiplication of 
a numeral set. But it is necessary to consider more closely what are the forms 
of those conditions of detachment which have been above alluded to, and which 
(according to the view here taken) are required for the (separate) existence of 
such a numeral set; it will also be proper to give, at least, some examples of the 
possibility of satisfying the conditions thus determined. 
Conditions of Detachment. 
18. The following appears to be a sufficiently simple mode of discovering the 
conditions of detachment, under which the values of the numerical coefficients, 
2,» 1 (149) or (150), shall be independent of the ratios of the ordinal con- 
stituents of the set q, which is originally operated upon. Employing the charac- 
teristics of ordinal separation, as explained ina former article, we may now regard 
it as being the definition of the sign of derivation X,, that this sign satisfies the 
symbolic equation, 
eee aS (156) 
which gives 
By Nr aires ee Ny 
Sy Cn Ona Bis (157) 
On the other hand, the equation (153), when operated on by the characteristic 
of separation R,, gives, by changing 7” to s, and by afterwards changing 7, s in 
(156) to s,s’: 
Ry Xp Xp = By, - Myr, Ry Xz 
PS ge nis Cp re Bie (158) 
We are then to satisfy the equation, 
= 5, .( 1} (BK Ce Ba Xr) 
= Let (Cee Cy et a Cy, eye Ox.) Ry3 (159) 
and because we are to do this independently of the ratios of the » constituent 
ordinal relations a,, which are obtained from the ordinal set q by the m operations 
