1896 - 97 .] Prof. Tait on Equations in Vector Functions. 499 
The members, besides being equal, are conjugates ; so that they 
represent any pure strain whatever. 
Thus x = and ^ = 7rr<^' ,_] , which are of course consistent 
with one another. Remark that, as a particular case, nr may be 
a mere number. If oT be taken = <£<£', we have the obvious 
solution x = <£• 
4. If we alter the order of the factors on one side of (2) we have 
an altogether new form : — 
Since cfi is given, this may be written 
where i p is known. An immediate transformation by taking the 
conjugate gives 
a type which is obviously a particular case of (T) ; and, besides, 
will be treated later, with the sole difference that x will then be 
the given function, and if/ that to be found. But when a solution 
has thus been obtained, it must be tested in the original equation ; 
for selective eliminations, such as that just given, often introduce 
irrelevant solutions. (See § 8, below.) 
5. A curious modification of (3) is produced by making in it 
and x identical, so that it becomes 
Though no longer linear, this equation is in some respects analogous 
to (1). It thus imposes the condition that x and its conjugate 
shall have the same directional roots. If all three be real they 
must therefore form a rectangular system. If two be imaginary, 
the vectors of their real and imaginary parts form a rectangular 
system with the third. Thus x ma y be any pure strain, or a 
rotation associated with a pure strain symmetrical about the axis of 
the rotation. 
A simpler mode of dealing with (4) is suggested by the last 
remark. For we may always assume 
(3). 
xx = xx 
/ / 
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