174 Proceedings of the Royal Society of Edinburgh. [Sess, 
where X and are constant vectors. This equality must hold for all 
values of p. Putting p = y gives (because 0y = ^y) 
SAy = 0, S/xy = 0 ; (20) 
and operating on both sides of (19) hy (p — g gives 
{<p-gfp^{cp-g)p^Xp, (21) 
because (^— ^)y = 0. The left side of (21) is parallel to y, in virtue of 
(18); hence the right side also. That is to say, the vector ((p—g)/3 is 
parallel to y. Accordingly, if we put p = /3 in the identity (19), the first 
term on the right must vanish, i.e. 
= 0 (22) 
By (20) and (22), X is at right angles to both /3 and y, and we may write 
\ = cV^y (23) 
Also, p being at right angles to y, must be of the form c-^Vya, where a 
is some constant vector, and a scalar. Substituting in (19) the values 
of X and /x thus obtained, 
(f>p = gp + c^Sf^yp + C-^ySyap . . . . (24) 
is a normal form for a linear vector function having all its axes 
coincident. 
Since the only limitation imposed upon has been the coincidence 
of roots of the symbolic cubic, we may infer that (24) contains, implicitly, 
seven scalar constants : the axis counts as two, the root g as one ; the 
precessive plane must contain the axis, but its aspect is otherwise arbitrary, 
and its angle with a fixed plane through the axis may be taken as a fourth 
scalar ; if we choose /3 as a unit vector making a fixed angle with the axis, 
and a a unit vector at right angles to the axis, the constants c and c^ are 
arbitrary; and, finally, the angle between a and ^ may count as the 
seventh scalar. 
To illustrate this strain by a physical example, let us suppose, as before, 
that a parallelepiped stands on a level table. Take ^ and y along the 
horizontal edges, and a along the non-horizontal edges. Apply in 
succession the two shears, 
ifp = p + a^S/3yp and a>p = p + a^ySyap, . . . (25) 
which by compounding give 
^p = coi/^p = p -f- (a/3 + aajySa/3y)S/3yp + a^ySyap, . . . (26) 
which is in the form (24). If we wish to obtain the most general strain 
