1896 — 97 .] Prof. Tait on Equations in Vector Functions. 501 
unless we refer the strain to the axes of its pure part ?*y, when it 
becomes fairly simple. For </> can then he written as 
( A 2 -v /x ) 
1 / B 2 - A 
- [x A C 2 
whence it is easy to see that 
X = (e + (1 - e)l' 2 - nf+( 1 - e)lm) 
B 
hnf+(l-e)lm) e + (l-e)m 2 
A 
( - mf + ( 1 - e)ln) ?( If + ( 1 - e)mn) 
A B 
^(m/+ (1 - e)ln) 
5( - //+ (1 - e)mn) 
V 
e + (1 - e)n 2 
where 
l = AX/ Jm? + B V + C V, etc., and 
e 2 +/ 2 =l . 
) 
7. A similar mode of treatment can, of course, be applied to the 
more general form 
X<t>X= x I / • ( 6 ). 
After what has just been said, it is easy to see that if f = zs x + Vej , 
we shall have 
X = t7rpox7r*, 
with the condition for to (and for tlie possibility of a solution) 
7nu)To*e = T3 , 
where m is the product of the numerical roots of x • 
[In connection with the results above it may be interesting to 
find the relations among the various constituents of the two 
different modes of breaking up a linear vector function into pure 
and rotational parts: — i.e. 
<f) = i rr + Ve = £r lW . 
(See Proc. R. S. E., vii. 316, for another solution.) 
The general form of a pure rotation is 
w = a Aln ( )a~ Alrr = cos A + sin A Va - (1 - cos A)aSa 
where a is the unit vector axis and A the angle of rotation. 
Thus, writing for shortness c — cos A and s — sin A, 
Top + Yep = cTTT^p + soj^Y ap — (1 — c)tH' 1 aSap 
Top - Yep = cTo } p - sY aTo ] p — (1 — c)aSa<77jp 
