178 Proceedings of the Eojal Society of Edinburgh. [Sess. 
where x-^, y^, and 2 ^^ are to be determined. We have 
cf>6a = 0(fia, if the strains are to be commutative, 
= 0{ga + c'[3), by (24), letting c =cSaj3y, 
= g{x^a + + z^y) + c\lll5 + zy\ 
by the conditions already proved for 0^, Again, 
cfiOa = (f>{x-^a + y-^/3 + Z^y) 
= x^(ga + c'/?) + ?j^{g(3 + c^y) + z^gy, by (24). 
By comparison of coefficients, 
Xi = h, cyy^^cz. ..... (35) 
These conditions serve to determine Oa in part, if we suppose d/3 previously 
determined by the assignment of values to h and to 0 , but they leave the 
y-component of da wholly arbitrary. We have, therefore, the rule : Two 
strains of class III are commutative when, and only when, they have 
the same axis and the same precessive plane, and satisfy the condition 
If we assume <p of class III, the conditions 
$a = ha + — zj3 + 
^1 
(9/3 = 7/^ + zy, 
Oy — hy 
determine the most general d commutative with cp. We may have d 
reducible when 0 = 0. 
12. Finally, if both cp and d are of class IV, we may assume 
<pp = gp + aSXpf Op = hp + (SSfxp. 
The requirement (p0p = 0^p, expanded, becomes 
ghp + g/3Spp + /iaSAp + aSA^Sp-p = ghp + haSXp + p/3Spp + ^SpaSAp, 
<• 
which simplifies at once to 
aSA/3Spp = /3SpaSAp. .... (37) 
It is evident that we must distinguish two cases. First, if a and /3 are 
parallel, we may also have A and p parallel. Second, we may have SX/3 
and Spa both zero, when a and /3 may have any direction. In the 
first case the functions have the same axes. In the second case, however, 
we have a much wider range, including, for example, the \p' and x of (1^) 
and (17) as special forms. The functions need not be shears (for which 
the conditions would be SAa and Sp/3 respectively zero). In physical 
language, we may say that two reducible linear vector functions are com- 
mutative if all axes of one are axes of the other; but if they have not 
