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-coujugate 
function, we have thus either 
<p = l Z' + V. € ( ), 
P = ) q~\ or <p = 5 .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 if/ and w each involve six, and e and 
q each three. If <p r be the function conjugate to <p, we have 
<p'= if/ - Y. € ( ) 
so that 
and 
2 = <p + f 
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. 0-1) where 0 is 
the angle of rotation. 
In the second form (whose solution does not appear to have been 
attempted) we have 
P = 2*( ) 2 —1 ! 
where the pure strain precedes the rotation; and from this 
<p'=^.q- 1( ) q, 
or in the conjugate strain the rotation (reversed) is followed by the 
pure strain. From these 
p'p = 1 (g'S ( ) q~ l ) q 
— Sr 2 , 
VOL. VII. 
4 T* 
