Sir Wiriram Rowan Hamiuton’s Researches respecting Quaternions. 201 
mary, secondary with secondary, and soon: thus if, with the recent significations 
of the symbols, we write the guaternion equation, 
/ 
Vg =a (3) 
or more fully, 
(Ap Ais Sas Aa) us CAyswAys es aus (4) 
we indicate concisely, thereby, the system of the four following momental equa- 
tions, or expressions of four coincidences between moments of time denoted by 
different symbols : 
A= Alege Ay ea A =A, At Ae (5) 
The same complex equation, or system of equations, may also be thus written : 
(Ca Bis Abs As) a5, Sy, =, =) if ayy Ant A An) (6) 
or more concisely thus : 
obG= 55 =), =), a. (7) 
Characteristics of momental Separation, Recombination, and Transposition. 
2. In the foregoing article, parentheses have been used as characteristics of 
systematic combination, in order to combine the symbols of separate moments 
into the symbol of a common set. If we now agree to prefix, conversely, charac- 
teristics of momental separation, such as M,, M,, . . . to the symbol of a momental 
set, in order to form separate symbols for the separate moments of that set, we 
may resolve the equation (1) into the four following : 
MQ A, 5) | \— "A 5) MQ — A, 5 “MiQ'== A, 5 (8) 
and an equation, such as (3), between two momental quaternions or other sets, 
Q and Q, may, in like manner, be resolved into equations between moments as 
follows : 
M,Q@ =M,@; MQ =m; &c. (9) 
With these characteristics of combination and separation of moments we may 
write, for any four moments, A, B, C, D, the tdentical equations, 
A= M, (A, B,C, D); B= M,(A,B,C,D); &c. (10) 
and for any momental quaternion q, the identity, 
Q= (M,Q, M,Q, MQ, M,Q) ; (11) 
with other similar expressions for other sets of moments. 
2E2 
