354 Proceedings of the Royal Society of Edinburgh. [Sess. 
for we cannot have simultaneously <py = 0 and <p 2 \ = y \ 2 ; this, however, 
may be regarded as that limiting case of (2) where both roots of co vanish. 
The imaginary case A 2 = 0 is easily examined by the foregoing methods, 
and may be left to the curiosity of the reader. 
Finally, if w is a mere dilation, i.e. if co=g, we clearly have an 
infinite number of values of <p, by assigning (pa = + Jg a, <pfi= — Jg /3, 
(py= ± Jg y, with Sa/3y not zero*; also when g = 0 we have solutions 
of the form <pp = /3S/3yp. 
The more important of the foregoing special cases are easily summed 
up. By inspection of the various cases we have 
Theorem III. — Solutions of the equation (p 2 = co fail to exist when, and 
only when, co has a double, or a triple axis with vanishing latent root.f 
Theorem IV. — If a linear vector function has an infinite number of 
axes, it has an infinite number of square roots ; except as provided in 
Theorem III. 
Theorem V. — A linear vector function having all its latent roots 
positive has always one, and only one, square root having its latent 
roots all positive. 
The methods by which these results have been obtained can be applied 
to cube roots and higher roots of a linear vector function, and the cor- 
responding theorems differ but little from those for square roots. The 
process consists essentially in writing co in normal form, assuming a <p 
with undetermined coefficients, raising <p to the proper power by direct 
operation, and equating to the known form of co. 
More generally, if <p satisfies the equation with scalar coefficients 
a n (f) n + a n _ 1 (f) n ~ 1 . . . +a 0 = (j) 
(p and co are commutative and the same method of attack might be used. 
But to solve a quadratic we should more naturally proceed thus : given 
a(p 2 + b<p + c = co, we complete the square on the left just as in ordinary 
algebra, then extract the square root, by the rules of the present paper, or 
otherwise. The more general quadratic, Pp(p = q, where p and q are 
* Comparing our results with Tait’s solution (note 1), with which, of course, they are 
fully in accord, it is interesting to note that when has an infinite number of values, the 
denominator w+g l in Tait’s solution becomes w— g, hence (oo + g 1 )~ 1 is indeterminate. 
y General conditions for the existence of a solution of the equation in matrices A 
are given by H. Kreis, Vierteljahrschrift Nat. Ges. Zurich, 1908, liii, p. 375. While his 
results hold for matrices of any order, their form renders them needlessly cumbrous for 
the present case. Cayley, by a method akin to Tait’s in using the symbolic cubics of the 
known and unknown matrices, showed how the square root can be extracted, considering 
the general case (eight solutions) only. Proc. Roy. Soc. Edin., 1872, vii, p. 675. 
