1914-15.] The Commutative Law for Homogeneous Strains. 179 
the same axes, and neither of them is a shear, the plane of indeterminate 
axes of each must include the single, isolated, axis of the other ; while if 
either function is a shear, the direction of shearing must lie in the plane 
of indeterminate axes of the other function. 
As a simple deduction, any number of shears having the same direction 
of shearing are commutative, although their axis planes are different. 
That is, if we take <pp = p-\- aSXp, and give X any number of values satisfying 
SXa = 0, while a is fixed, the resulting values of 0 are commutative, while 
the axis plane, in each case, is the plane perpendicular to X. Thus the rule 
for coincidence of axes does not hold for two strains of class IV. And we 
may, with equal simplicity, keep X fixed, varying a under the condition 
SXa = 0 ; the resulting strains will be commutative and will have their 
axial planes in coincidence, but the double axes a are different by 
hypothesis ; each such strain has all planes through its own particular 
value of a precessive ; hence the rule for coincidence of precessive planes 
does not hold for two strains of Class IV. In fact, (37) is both necessary 
and sufficient for this case. 
13. Illustrations of the commutative properties of strains might be 
given almost without number. For example, in solving equations in 
linear vector functions (of which Tait gives several sets), the ability to 
interchange the order of two functions is a prime necessity. As an 
instance of a new method of attack for an old problem, let it be required 
to integrate the partial differential equation 
ScrVw = 0, ...... (38) 
where o- is a given vector — a function of p, and u is to be found. One 
method of solution is to determine a vector tt which shall be normal to 
a family of surfaces (so that tt is parallel to Vu), and also perpendicular 
everywhere to <j, so that S(rVi^' = 0. By writing 7t = V( 7 t, we have tt 
perpendicular to cr. That tt shall also be normal to a family of surfaces 
we must have SttVtt^O, or 
0 = So'rV(crr) = S(tt(tSVo’ — crSVr + — Ocr) * 
= So-r(<^T — Ocr), ...... (39) 
(p and d being the linear vector functions obtained by differentiating a- 
and T, respectively. If we now take or and r homogeneous of degree n 
in p, (pp = n(T, and Op = nr, it is evident that (39) becomes 
S(TT{(f)6 — 0(f>)p = 0, ..... (40) 
whence one way of finding a value of r, if a- is homogeneous, is to find 
an integrable Odp with <p and 0 commutative. If n=l, that is, if (r = (/)p, 
* Phil. Mag., June 1902, p. 579. 
