200 Sw Writriam Rowan Haminton’s Researches respecting Quaternions. 
to the Academy, since the date prefixed to this First Series of Researches, will 
tend to place the subject in a still clearer point of view: and, by exhibiting more 
fully to mathematicians its interest and its importance, increase the likelihood of 
their contributing their aid to its development. 
Observatory of Trinity College, Dublin, May 3, 1847. 
General Conception and Notation of a System or Set of Moments. 
1. When we have in any manner been led to form successively the separate 
conceptions of any number of moments of time, we may afterwards form the new 
conception of a system, or MOMENTAL SET, to which all these separate moments 
belong; and may say that this set is of the second, third, fourth, or n™ order, 
according as the number of the moments which compose it is 2, 3, 4, or m: we 
may also call those moments the constituent moments of the set. A symbol for 
such a set may be formed by enclosing in parentheses, with commas interposed 
between them, the separate symbols of the moments which compose the set; thus 
the symbol of a momental quaternion, or set of the fourth order, will be of the 
form 
(46s Ay Ags Ay)s 
if Ay, Aj Ay, A, be employed as symbols to denote the four separate moments of 
the quaternion. If we employ any other symbol, such as the letter 9, to denote 
the same quaternion, or set, we may then write an equation between the two 
equisignificant symbols, as follows : 
Qu=—(Ar, Aa aas AG) (1) 
and, in like manner, if q’ denote another quaternion, of which the four separate 
moments are denoted by aj, aj, Aj, 4s, we shall have this other similar equation, 
cay (i A AS) (2) 
An equation of this sort, between two symbols of equinumerous momental sets, 
is to be understood as expressing that the several moments of the one set coincide 
respectively with the homologous momeits of the other set, primary with pri- 
