350 
Proceedings of the Royal Society of Edinburgh. [Sess. 
XX. — The Square Roots of a Linear Vector Function. By Frank 
L. Hitchcock. Communicated by The General Secretary. 
(MS. received February 26, 1917. Read May 19, 1917.) 
The equation in linear vector functions 
</> 2 = w . . . . . . . ( 1 ) 
was proposed by Tait,* and an elegant solution was obtained by him which 
does not require a determination of the axes of go. He showed that upon 
this equation depends the separation of the pure and the rotational parts 
of a homogeneous strain. The problem appears to be interesting also from 
the point of view of algebraic analysis. The number and character of the 
solutions is more varied, given different types of the function go, than 
we might at first suppose. In fact there are two forms which may be 
assigned to go sucli that the equation does not permit of solution. Other- 
wise the number of solutions is 2, 4, 8, or infinity. Keeping Tait’s 
notation for the cubic in go, the two cases of failure arise when m — 0 ; 
hence the solution is always possible for a physically existent strain go. 
But there are also cases where m = 0, and the solution exists. Further- 
more, Tait's solution ceases to be determinate when go has an infinite 
number of axes — for example, when go is real and self-conjugate with a 
pair of equal roots. 
In a former paper were given four normal or type expressions covering 
all cases in the sense that a given go can always be thrown into one of the 
four forms. f In the same paper it was proved that if two linear vector 
functions are commutative, and if the first has an axis not an axis of the 
other, then the first is reducible, i.e. possesses an infinite number of axes. 
Now if <p 2 = oo, it is obvious that go and <p are commutative, since any number 
of linear vector functions are associative. Hence we have the theorem — 
Theorem I . — A linear vector function is reducible if it has an axis not 
possessed by its square root. 
The root (p, on the other hand, can evidently not have an axis which 
is not an axis of (p 2 . Then if cp 2 be assigned to have a finite number of 
axes, <p has those, and no others. Whence this second theorem — 
* Quaternions , 3rd ed., cliap. v, Ex. 17 ( a ), p. 144 ; solution, p. 298, note. Also in 
Kelland and Tait’s Introduction to Quaternions , chap. x. 
f Proc. Roy. Soc. Edin., vol. xxxv, Part ii (No. 17), p. 170 (1915). 
