Sir Wiit1am Rowan Hamirton’s Researches respecting Quaternions. 239 
But, also, 
NO eS SU Nilo Se SS (212) 
and, therefore, 
Ni OSD) SAT TER (213) 
consequently, by operating with the characteristic n, on the symbolical equation 
(205), we obtain this other form for the expression of the associative principle, 
considered as establishing a certain system of relations between the coefficients of 
multiplication : 
OS 2 (OAs ae ae) (214) 
We are, therefore, entitled to regard this last formula, or the system of numeri- 
cal equations of condition which it includes, as being a consequence of the analo- 
gous system of conditions included in the formula (160), because the associative 
property of multiplication is a consequence of the principle of detachment. And 
on comparing the two formulz, we perceive that as soon as the one last deduced, 
namely, (214), has been satisfied by a suitable system of coefficients of multipli- 
cation, then the one previously established, namely, (160), can be immediately 
satisfied also, by connecting with this latter system asystem of coefficients of deri- 
vation, according to the rule expressed by the following very simple equation : 
Ce i esas (215) 
For example, in the case of couples, with the abridged symbols (163), (164), 
for the two systems of coefficients, this rule (215) would have shewn that if we 
had in any manner succeeded in satisfying the sixteen equations of detachment 
(165), ...(168) between abed a’b‘c'd’ and efgh e’f'g’h', we could then satisfy the 
same equations of detachment with the same values of the eight latter symbols, 
and with the following values for the eight former : 
ine oh. (bi She! oe es=aifs, domenf' 5 | (216) 
A peep ahs ad Sh"'; 
which, in fact, will be found to agree with the values of the nineteeth article. 
Connexion between the Coefficients of Derivation and of Multiplication ; sim- 
plified Conception of a numeral Set, regarded as expressing the complex 
Ratio of an ordinal Set to a single ordinal Relation. 
22. The rule (215), for connecting together the two systems of coefficients, 
VOL. XXI. 2k 
