1896 - 97 .] Prof. Tait on Equations in Vector Functions. 497 
Note on the Solution of Equations in Linear and Vector 
Functions. By Prof. Tait. 
(Read June 7, 1897.) 
In a paper read to the Society on March 1 (ante, p. 310) I spoke 
of the application of some of its results to the solution of equa- 
tions involving an unknown Linear and Vector Function. These 
results depended chiefly upon the expression of the function in 
terms of its roots, scalar and directional ; and I now give a few 
instances of their utility, keeping in view rather variety of treat- 
ment than complexity of subject. The matter admits of practically 
infinite development, even when we keep to very simple forms of 
equation, and is thus specially qualified to show the richness in 
resources which is so characteristic of quaternions. But it will be 
seen also to be strongly suggestive of the extreme caution re- 
quired even in the most elementary parts of this field of inquiry. 
In what follows, 1 employ ^ to denote the unknown function ; 
<f>, i (f, etc., known functions. 6T is specially reserved for a self- 
conjugate function, and <o for a pure rotation. 
1. Given <f>X = X<t>\ .... (1), 
i.e., to find the condition that two functions shall be commutative 
in their successive application. Let a be a root of </>, real or 
imaginary, so that 
cj) a = g a . 
We have at once, by applying the members of the proposed 
equation to a, 
^ X a = X ctm = gx a • 
Thus, except in the case of equal roots of <f > , 
= ha ] 
so that the required condition is merely that y has the same 
directional roots as <f>. When two values of g are equal, two of 
the directional roots of y are limited only to lie in a definite 
