212 Sir Woxtam Rowan Hamitton’s Researches respecting Quaternions. 
foundation of the theory of algebraic couples, and of the use of the symbol Y —1 
in algebra, proposed by the present writer, in that Essay, already several times 
referred to, which was published in a former volume of the Transactions of this 
Academy; for the symbolic equation (vol. xvii. page 417, equation 157) 
V(—1) = (0, 1), 
was there given, in which the essential character of the nwmber-couple (0, 1) 
was that, when used as a multiplier, it transformed one step-couple (a, a,), that 
is to say, one couple of steps, a,, a,, in the progression of time, or one couple of 
ordinal relations between moments, into another couple of steps or of relations in 
the same progression of time, according to the law, 
(0, 1) (a, a, ) = (= ay a) > 
which agrees with the process directed by the recent characteristic of derivation, 
R_,,9, and was included in the equation (37), page 401, of the volume lately 
cited. Again, if we now regard 7, 7, & as three characteristics of operation on an 
ordinal quaternion, defined as follows: 
9 =p eet 5 | 
J = B_3,0, -13 i (70) 
— R_3, -9,1,03 
we shall have the four following symbolic equations, which will be found to be 
of essential importance in the present theory of quaternions : 
ale 
(pS | is 
Koes (71) 
igi =— 1; 
and which may be concisely expressed under the form of a single but continued 
equation, as follows : 
RSP Se SYS = 1. (72) =(a) 
7. To leave no doubt respecting the truth or meaning of these important 
symbolical relations, (72) or (a), between the three coordinate characteristics of 
quaternion-derivation, i, j, k, defined by the equations (70), we shall here ex- 
hibit distinctly the successive steps or stages of the transformations which are 
