1916-17.] The Square Roots of a Linear Vector Function. 351 
Theorem II . — A given linear vector function and its square root are of 
the same class except when the given function is reducible. 
If oo is reducible, it will be shown that we have, in general, an infinite 
number of irreducible values for <p, as well as a finite number of reducible 
values. 
Considering in order the possible forms of oo, if co is of Class I, i.e. if 
oo has just three axes, the matter is simple, for cp is fully determined by 
possessing as axes the axes of co, and as latent roots the square roots of 
those of co, each taken with either sign. Hence there are eight solutions, 
except when m = 0, i.e. when co has a vanishing latent root, the number 
of distinct values of (p falling to four. 
When co is of Class II, i.e. has but two distinct axes, we may write 
M p = g p + C/3S/3/V + C^Sa/fy .... (2) 
where c, c v and S af3/3 1 do not vanish. f3 is the double axis, g the double 
root, /3 1 and g -{-c 1 Sa/3^ l the single axis and root. The plane of a and /3 
is precessive, i.e. repeated application of co to any vector in that plane 
brings it more and more into the direction /3. Now, as above noted, <p 
has /3 and /3 1 for axes, and no others. The latent roots of cp corresponding 
to these axes must be the square roots of those of co, taken with either 
sign, four possibilities. It only remains to find the effect of cp on a. Let 
The latent roots of cp be selected and called r and r v Assume 
<pa = xa + y/3 + zf3 v 
-where x , y, z are to be determined. Operating again with <p , we have 
<£ 2 a — X<pa + y<p/3 + zpf3 1 
= x 2 a + xyj 3 + xzf3 1 
+ ry(3 + r 1 zf3 1 
by the values of (pa, <p{3, and (p/3 1 just assumed. But by (2) 
<p 2 a --- ohx = ga + c/3 Sa/3/3 15 
whence equating values of (p 2 a 
x 2 — g , y(x + r) = cSa/3f3 1 , z(x + ?q) = 0. 
•By the first of these equations x is one of the square roots of g. By the 
second equation x cannot be the negative of that square root of g already 
selected to be r, for if so either c or S a6/3 1 would vanish, contrary to 
hypothesis. Hence x — r. Thus y is determined. Since two latent roots 
-of co are unequal, we cannot have x J r r 1 = 0; hence z = 0. Thus <pa is 
.uniquely determined when r and r ± have been selected. We therefore 
