202 Sir Wiru1am Rowan Hamirton’s Researches respecting Quaternions. 
The identical expression (11) may also conveniently be written thus : 
TQ = Gap a, Masi) Ola es; (12) 
1 g being regarded as a symbol equivalent to @, and the third member of the for- 
mula being an abridgment of the second ; and then, by omitting the symbol q of 
that quaternion of moments which is here the common operand, we may write, 
more concisely, 
= (Mp Mis My My) Mo. a5'8 (13) 
and may call the second or the third member of this last symbolical equation a 
characteristic of recombination (of a momental set). The same analogy of 
notation enables us easily to form characteristics of momental transposition, 
which shall serve to express the effect of changing the places or ranks, as primary, 
secondary, &c., of the moments of any set, with reference merely to that con- 
ceived and written arrangement on which the set itself depends for its subjective 
_ or symbolic existence, and without any regard being here had to the objective or 
phenomenal succession of the moments in the actual progression of time. Thus, 
from the proposed or assumed quaternion (1), we may, in general, derive twenty- 
three other quaternions, which shall be all different from it, and from each other, 
in consequence of their involving different mental and symbolic arrangements of 
the same four moments of time; and these new quaternions may be denoted by 
the following expressions : 
(Ay Ay Ag Ay) = Mo,1,3,293 
eae (14) 
(Ruy Ns Bp By) = Shame 
In this notation we may write the symbolical equations, 
Me oes eM s bes (15) 
to imply that four successive transpositions, which are each of the kind directed 
by the characteristic M,,,,,., will reproduce any proposed momental quaternion 
(A, B, C, D), as the last of the four successive results : 
(B54; 8, GC); (CG, Dy ASB)y e(ey Ge, Dia), Mas Bec, D))- (16) 
And generally, for any set of moments, we may write, by an analogous use of 
exponents, the formula 
i 
Meo Mo nea e les (17) 
