of Edinburgh, Session 1871-72, 
667 
3. Note on the Strain-Function. By Professor Tait. 
When the linear and vector function expressing a strain is self- 
conjugate the strain is pure. When it is not self-conjugate, it may be 
broken up into pure and rotational parts in various ways (analogous 
to the separation of a quaternion into the sum of a scalar and a vec- 
tor part, or into the 'product of a tensor and a versor part), of which 
two are particularly noticeable. Denoting by a bar a self-conjugate 
function, we have thus either 
9 = if/ + V. e( ), 
p = 2 S( ) q~\ or f> = 5 .j ( )q- 1 , 
where e is a vector, and q a quaternion (which may obviously be 
regarded as a mere versor). 
That this is possible is seen from the fact that <p involves nine 
independent constants, while ^ and w each involve six, and e and 
q each three. If <p' be the function conjugate to <p t we have 
<p'= ^ - Y. € ( ) 
so that 
and 
2if/ = <p + <p' 
2 Y. e ( ) = <P - <p' 
which completely determine the first decomposition. This is, of 
course, perfectly well known in quaternions, but it does not seem 
to have been noticed as a theorem in the kinematics of strains that 
there is always one, and but one, mode of resolving a strain into the 
geometrical composition of the separate effects of (1) a pure strain, 
and (2) a rotation accompanied by uniform dilatation perpendicular 
to its axis, the dilatation being measured by (sec. 6-1) where 6 is 
the angle of rotation. 
In the second form (whose solution does not appear to have been 
attempted) we have 
P = ( )2 -1 , 
where the pure strain precedes the rotation ; and from this 
P'= 5 -2~ 1 ( ) 1 > 
or in the conjugate strain the rotation (reversed) is followed by the 
pure strain. From these 
P'P = (?» ( ) 2 — J ) 1 
_ -2 
4 T* 
VOL. VII. 
