1895-96.] Prof. Tait on the Linear and Vector Function. 161 
But (1) may be written in the form 
o r 1 'Gyp = gp, 
where os~ 1 w is in general not a self-conjugate function. Thus 
Two pure strains in succession give a strain which is generally 
rotational, but whose cubic has three real roots. 
Conversely, when a strain is such as to leave unchanged three 
directions in a body, it may be regarded as the resultant of two 
successive pure strains. 
These are to be found from (2), in which the values of g and p 
are now regarded as given, so that the problem is reduced to find- 
ing o> (a pure strain), and the (rectangular) values of a from three 
equations of the form 
<Fp = 0 \ 
When co is thus found, the value of m is given by (1). The solu- 
tion is easily seen to express the fact that co and us, alike, convert 
the system p v p 2 , p 3 into vectors parallel to Vp 2 p 3 , V p 3 p v V/qp 2 , 
respectively. 
2. Other modes of solution of (1) are detailed, of which we 
need here mention only that which depends upon the formation of 
the cubic in 
— tv — g co, 
the calculation of the coefficients in M 0 and the comparison of 
these forms with their equals found from 
</> — nr co ~ 1 - g, 
and from </> = oT^co - * - g ; 
a process which gives interesting quaternion transformations. 
3. Some curious consequences can be deduced from these 
formulae, which have useful bearing upon the usual matrix mode 
of treating the problem algebraically. 
For, if we take 
( A 
c 
b ) and co = ( p 
0 
0 ) 
c 
B 
a 
1 ° 
<1 
0 j 
b 
a 
C 
I 0 
0 
r 1 
YOL. XXI. 
21 / 12 / 96 . 
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