300 
[836 
836. 
ON THE QUATERNION EQUATION qQ-Qq' = 0. 
[From the Messenger of Mathematics, vol. xiv. (1885), pp. 108—112.] 
I CONSIDER the equation qQ — Qq' = 0, where q, q are given quaternions, and Q 
is a quaternion to be determined. Obviously a condition must be satisfied by the 
given quaternions; for, substituting in the given equation for q, q', Q their values, say 
w + ix+jy + kz, iv + ix +jy + kz, and W + iX + j Y + kZ respectively, and equating to 
zero the scalar part and the coefficients of i, j, k, we have four equations linear in 
W, X, Y, Z, and then eliminating these quantities, we have the condition in question. 
Supposing the condition satisfied, the ratios of W, X, Y, Z are then completely determ 
ined, and the required quaternion Q is thus determinate except as to a scalar factor, 
or say Q is = product of an arbitrary scalar into a determinate quaternion expression. 
It might, at first sight, appear that the condition is that the given quaternions 
shall have their tensors equal, Tq — Tq ; for the equation gives Tq.TQ — TQ. Tq' = 0, 
that is, TQ(Tq—Tq') = 0. But we cannot thence infer, and it is not true, that the 
condition is Tq — Tq' = 0; the formula does not give the required condition at all, but 
the conclusion to which it leads is that, when the condition is satisfied, then in general 
(that is, unless Tq — Tq = 0) the required quaternion is an imaginary quaternion (or, 
as Hamilton calls it, a biquaternion) having its tensor TQ = 0. In the particular case 
where the given quaternions are such that Tq - Tq = 0, then the required quaternion 
Q is determined less definitely, viz. it becomes = product of an arbitrary scalar into 
a not completely determined quaternion expression; and it is thus in general such 
that TQ is not = 0. In explanation, observe that, for the particular case in question, 
the four linear equations for W, X, Y, Z reduce themselves to two independent rela 
tions, and they give therefore for the ratios of W, X, Y, Z expressions involving an 
arbitrary parameter A; these expressions cannot, it is clear, be deduced from the 
determinate expressions which belong to the general case.