1914-15.] The Commutative Law for Homogeneous Strains. 171 
If p be distinct from the other axes, at least two of the coefficients 
x^, x^, x.^ must be different from zero. Suppose x-^ and x^ not zero. 
Equating coefficients of /3-^ and /S^ in (2) and (3), we have g-^ equal to g^. 
The strain in the plane of and is therefore a uniform dilation, and 
every vector in that plane is an axis. 
2. Strains of the general type, having three, and only three, distinct 
axes, have been clearly described.* They may always be written in the 
form (2). If we multiply both sides of (1) by take the scalar part 
of the product, we find 
(I) 
__ and similarly x - 
We may, if we prefer, replace these scalar products of three vectors by the 
determinant of their components along axes of co-ordinates. The case of 
imaginary values for a pair of g's has also been sufficiently treated.f 
3. To obtain the most general strain with a double axis, let cp convert 
/3 into g^ and into Suppose ^ to be a double root of Hamilton’s 
symbolic cubic,J that is, 
(5) 
identically. In consequence, (0 is annulled hy {(p — g-^). What is the 
same thing, {(p — gYp is a vector parallel to /5p or else null. Moreover, 
{(p—gYp is linear in p. In any case, therefore, we must have 
{cp-gy^p = (3^SXp, ( 6 ) 
where A is some constant vector (perhaps null), because any scalar linear 
in p can be written SAp. 
By similar reasoning, {<p—gi)((p—g)p is parallel to /3, and 
= ( 7 ) 
where p. is another constant vector (or null). Subtracting (6) from (7), 
the second power of <p cancels, the remaining expression on the left factors, 
yielding 
(^-.di)(<^ (8) 
Assuming g different from we may now divide and transpose, and find, 
for the most general form of (p having a double axis, 
</)p = ^p + /5Sp'p + p\SA'p, ..... (9) 
A' and p,' being scalar multiples of A and p. The equality (9) must hold 
for all values of p. Putting p= /3 gives (because (p/3 = gl3) 
SA'/3 = 0, = (10) 
* Kelland and Tait, loc. cit. 
I Lectures on Quaternions, (7), p. 567. 
t Tait, Quaternions, 3rd ed., art. 176. 
