214 Sir Wit1iam Rowan Hamixron’s Researches respecting Quaternions. 
so that this repeated process of derivation by the characteristic i has changed the 
sign of the quaternion, q, by changing the sign of each of its four constituent 
ordinal relations, a, b, c,d; which is the property expressed by the first equation 
(71), namely, by the formula, 
e?= 1. (71, 1) 
By exactly similar operations, except so far as the second symbolic equation 
(70) differs from the first, we find, for the second coordinate derivative, jq, of 
the same proposed quaternion, q, the expression, 
jg = (— © d, a, — b); (80) 
and for the derivative of the derivative in the second mode, 
PU =IIV=(—a —b, —G —d)=—q=—lq; (81) 
the symbols 1q and q (like 1@ and Q) being regarded as equivalent: which re- 
sult (81) justifies the second equation (71), by giving the symbolic equation, 
f=. (Tl) 
And in like manner the third coordinate derivative, kq, is, by the third equation 
(70), expressed as follows : 
kq = (—d, —¢, b, a); (82) 
so that, by repeating this process of derivation, we find that the derivative of the 
second order, in the third mode, as well as in each of the two other modes, is the 
original quaternion with its sign changed, 
k’q = k.kqg = (—a, —b, —c, —d) = — 1q; (83) 
or, by detaching the symbols of operation from those of the common operand, 
e=—1. (71, 3) 
Finally, if we operate on the expression (82) for kq, by the characteristic 7, 
we find 
7-kG = Fs 30+1(— —, b; a) 
= (— b, a, —d,c) = 4q; (84) 
and, therefore, operating on this result by 7, we obtain, 
