1898-99.] Prof. Tait on Linear and Vector Function. 
547 
On the Linear and Vector Function. By Prof. Tait. 
(Read May 1, 1899.) 
(Abstract.) 
Three years ago I called the attention of the Society to the 
following theorem : — 
The resultant of two pure strains is a homogeneous strain which 
leaves three directions unchanged ; and conversely. 
[It will be shown below that any strain which has three real 
roots can also be looked on (in an infinite number of ways) as 
the resultant of two others which have the same property.] 
As I was anxious to introduce this proposition in my advanced 
class, where I was not justified in employing the extremely simple 
quaternion proof, I gave a number of different modes of demonstra- 
tion ; of which the most elementary was geometrical, and was based 
upon the almost obvious fact that 
If there be two concentric ellipsoids , determinate in form and 
position , one of which remains of constant magnitude , while the 
other may swell or contract without limit ; there are three stages at 
which they touch one another. 
[These are, of course, (1) and (2) when one is just wholly inside 
or just wholly outside the other (that is when their closed curves of 
intersection shrink into points), and (3) when their curves of inter- 
section intersect one another. The whole matter may obviously 
be simplified by first inflicting a pure strain on the two ellipsoids, 
such as to make one of them into a sphere, next considering their 
conditions of touching, and finally inflicting the reciprocal strain.] 
But the normal at any point of an ellipsoid is the direction into 
which the radius-vector of that point is turned by a pure strain ; so 
that for any two pure strains there are three directions which they 
alter alike. (These form, of course, the system of conjugate 
diameters common to the two ellipsoids.) This is the funda- 
mental proposition of the paper referred to, and the theorem 
follows from it directly. 
