1914-15.] The Commutative Law for Homogeneous Strains. 177 
Again, 
(f>ea = cf>(xa + tj^ + z[3^), by (30), 
= %a + c'/3) + yy^ + 2^ift, by (14) and (13). 
If we now compare values of (pOa, we find, from the terms in a, an identity ; 
the terms in ^ give cx = ch, and, as c is, by hypothesis, different from zero, 
we must have x — h; the terms in /3^ give gz = g^z, and as, by hypothesis, g is 
different from g-^, we must have z = 0. Accordingly y is arbitrary. But 
the conditions 
z = 0, x = h (31) 
are precisely the conditions that the precessive plane of (p should also be 
precessive for 0. We have therefore the rule: Two strains of class II 
are commutative when, and only when, they have the same axes and the 
same precessive plane. 
If we assume cp of class II, conditions (29) and (31) determine the 
most general form of d commutative with <p. We may have d reducible 
either when y = 0 ov when h = hy 
11. If we assume <p to be of class III, the problem is slightly more 
complex, but the same method may be applied to find the most general 
form of d so that 0(p = (p0. Let <p be thrown into the normal form (24). 
The axis of 0 must, by Theorem I, be an axis of d, that is, 
Oy = hy, (32) 
where is a constant scalar. Suppose 0/3 resolved along a, j3, and y, 
0(3 = xa + y(3 + zy, . . . . . (33) 
where x, y, and 0 are to be determined. We have 
cp0/S = 0<pl3, if the strains are to be commutative, 
= 0{g^ + c\y), by (24), letting c\ = c^Saf3y, 
= gxa + yy/3 + gzy + c\hy, by (32) and (33). 
Again, 
00/I = 0(a:a + y/I + 2 y), by (33), 
= x{ga + c'P) + y{g[3 + c\y) + zgy, by (24), with c = cSaf3y. 
Comparing coefficients of a, and y in the two expressions for (p0j3, we 
find, from terms in /3 (because c is, by hypothesis, different from zero), 
that X must be zero ; and from terms in y (because is not zero), that y 
must equal h. Whence the plane of (3 and y must be precessive for d as 
well as for cp. 
These conditions are, however, not sufiicient, for as yet no account 
has been taken of dct. Suppose 
Oa — x-^a + y^f^ + zyy, ..... (34) 
VOL. XXXV. 
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