1914-15.] The Commutative Law for Homogeneous Strains. 175 
of the type, we may follow i/r and w with a uniform dilation g, equal to 
the root of the cubic. (The root of the cubic for (26) is equal to unity.) 
It may be noticed that the and oo of (25) differ from \fr and ^ of 
Art. 4 in not being commutative. For 
if/dip = p + a^^ySyap a/3S/3yp, .... (27) 
which, however, is also of the type under discussion, having y for its only 
axis, with (py = y. 
6. It remains to consider strains with an infinite number of axes. A 
necessary condition for indeterminateness of axes is the identical vanishing 
of the product (<p — g)((p — g-^), where g and g^ are two roots of the cubic 
in <p. One of these roots must be multiple, for if all three roots are 
distinct we may write </)p as in (2). Suppose g to he a double root. The 
vector (<p — g)p must then be parallel to the axis of for all values of p 
for which it does not vanish — that is, 
4>P = 9P + y^^P^ ( 28 ) 
where y is the axis of g-^, and X is some constant vector (perhaps null). 
We may take (28) as a normal form for a linear vector function having 
an infinite number of axes. Any direction perpendicular to X is evidently 
an axis. There can, by the form of the expressions involved, be no other 
axis except y ; so that in case SXy = 0 all the axes are in one plane. If 
SXy = 0 and g = 1, the strain reduces to a shear. 
7. To sum up the discussion of homogeneous strains : Any linear vector 
function, hence a fortiori any homogeneous strain, may be expressed by 
one of these four normal forms : — 
I. y>p . S/3^/?2/?3 = + , 72 ^ 2 ^/^ 3 /^iP 9z(^fi*(^i(^2Pi 
where g^, g^, g^ are unequal roots of the 0-cubic. 
II. (f)p = gp + cf3Sl3/3^p + cfifiafSp, 
where g is a double root of the cubic, and may vanish. It is assumed 
c and do not vanish. Sa^p = 0 defines a precessive plane for this form 
of (p, not containing 
III. <pp — gp + (fiSfSyp + c^ySyap, 
where ^ is a triple root of the cubic, and may vanish. It is assumed 
c and and Sa/3y do not vanish. The plane of /3 and y is precessive. 
IV. <pp = gp + ySXp, 
where ^ is a double root with indeterminate axis. The vector X may 
vanish, when the strain becomes a uniform dilation. It will be convenient 
to speak of strains with an infinite number of axes as reducible. 
