1914-15.] The Commutative Law for Homogeneous Strains. 173 
shall be equal, we may infer that (14) contains, implicitly, eight scalar 
constants: the axes /3 and each count as two; the angle between the 
precessive plane and the plane of the axes as a fifth ; and the three scalars 
c, Cj, and g make eight. 
To illustrate how a strain of this type may arise, let a parallelepiped, 
on a level table, first undergo a shear parallel to one horizontal edge, then 
be uniformly stretched parallel to a different horizontal edge. The plane 
of the base is fixed. Take a parallel to the non-horizontal edge, /5 in 
direction of shearing, and in the direction of stretching. The shear * is 
P -H (16) 
which we may call \Jrp. The stretching is 
p + C-^f^lSa[3f, (17) 
which we may call yp. The final result <pp, that is obtained by 
writing \frp for p in (17), giving 
(f)p = p + c/lS^^jp + CjjdjSa^p, 
agreeing with (14). Both the shear and the stretching, considered 
separately, have an infinite number of axes. The axes of the shear are 
all horizontal. The axes of the stretching are the direction of stretching 
/5i, and any line in the face which cuts this direction. This latter face 
of the parallelepiped is a precessive plane for the shear and it is also 
the precessive plane for (p. 
5. To obtain the most general strain having a triple axis, we must 
assume the symbolic cubic to be a perfect cube. Let y be the axis, with 
fpy = gy. Then 
..... (18) 
identically. Consider now the vector {<p—g)p. Its locus must be a fixed 
plane, p being given all possible values. (This is a property of any strain 
(j> whatever, g being any root of the cubic, because the corresponding axis 
is reduced to zero by cp—g.) In the present case, by (18), the repeated 
operation (0 — pf)2 reduces every vector to the direction y. Therefore the 
axis lies in the plane of {<p — g)p, which is thus, by definition, a precessive 
plane. Let /3 be any other vector in this plane. Then 
(P - g)p^ P^^P + ySpp, (19) 
* Tail, Quaternions, 3rd ed., p. 299. The axes in Tail’s example are the axes of the 
pure part of the strain. 
