352 
Proceedings of the Royal Society of Edinburgh. 
have in general four distinct solutions of <p when go is of Class II. If the 
single root of go vanishes, the number falls to two, r ± being zero. 
If, however, the double root g vanishes, then <p ceases to exist for 
x — r — 0, and either c or S a/3/3 1 vanish, contrary to hypothesis. As a 
simple example of a function without square root, take Gop =jSip + JcSkp, 
of which the solution requires that <pj = 0 and <pH = —j, incompatible 
conditions. 
When go is of Class III, i.e. has but one distinct axis, we may write 
top = gp + cf3SfSyp + C^Syap ..... (3) 
where c, c 1 and Sa/3y do not vanish, y is the triple axis and g its latent 
root, y must be an axis of <p, and we select r, the latent root of <p, to be 
one of the square roots of g. Assume 
whence 
Again 
cpa = Xa + y'/3 + Zy , (p/3 = Xa + y/3 + Zy , 
<p' 2 (3 = X(pa 4- y(p[3 + zpy 
— xx a + xy'f3 + xzy 
+ xya + y 2 f3 + yzy 
+ rzy 
= 00/3 
g/3 + Cj^ ySa/?y, by (3). 
c p^a = X (pa + y'(p/3 + Z(py 
= x 2 a + x'y' (3 + xzy 
+ xya + yy'f3 + yzy 
+ rzy 
= ooa 
= ga +c/3Sa/3y, by (3). 
Equating coefficients, we have these two sets of equations, 
x(x' + y) = 0, xy' + i/ = g, xz + z(y + r) = tqSa/Sy, 
x' 2 + xy' = g, y'(x + y) — cSaf3y , yz + z\x + r) = 0. 
By the second equation of the lower set we cannot have x' + y^O, since c 
and Sa,6y are different from zero. Hence by the first equation of the 
upper set x = 0 . Then by the right hand of the upper set y+r is not 
zero; and y = r by. the second equation. Thus x, y, and 0 are uniquely 
determined when r has been selected. 
Writing x = 0 and y = r in the lower set, we have x' = r by similar 
reasoning. y' and z' are then uniquely determined by the last two 
equations. 
