252 Sir Witr1am Rowan Hamrtton’s Researches respecting Quaternions. 
detachment will have been satisfied, and a numeral se¢, of the kind above sup- 
posed, will be found under the form, 
q = Ly. MM, Xp = Ly Me % J (276) 
For the case of couples, we may make 
i eh when / (seo qun="Is (277) 
and then the condition (273) will be satisfied by the values of the coefficients of 
multiplication assigned in the nineteenth article; and the numeral couple will 
present itself under the well-known form, m, + m,/(—1). 
For the case of guaternions, if we suppose 
temas LPO rales PSP at Tete Bk Rael 9p (278) 
the symbols 7, 7, k being still connected by the fundamental relations (a); the 
six symbolical equations (228), and the sixteen symbolical equations (239), will 
then be included, by (269), in the formula (273), in which we may write, by 
(240), and by (271), or (238), 
qo = 4 bi + oj + dk; (279) 
and the expression (255) will be included in the more general expression (276). 
And if we farther particularize, and at the same time simplify, by adopting, as we 
propose henceforth to do, the values (241), which reduce q, to 1, we shall then 
obtain from (276), by (278), the same expression (254), or (c), which has already 
been assigned in the twenty-seventh article, as the representation of a numeral 
quaternion. 
Successive Multiplication of Quaternions: Application of the associative 
Principle. 
30. It has been stated that we design to adopt, in our theory of numeral 
quaternions, the simplifications contained in the equations (241). We shall there- 
fore regard, henceforth, the constituents of any product of éwo numeral quater- 
nions as being given by the simpler formule (248), and not by the more 
complex formule (260), in which A,...D, are abridged representatives of the 
sixteen quadrinomials (261) ...(264). Yet the trouble of investigating these 
latter expressions will not have been thrown away: for we may see, by (257), 
