170 Proceedings of the Royal Society of Edinburgh. [Sess. 
XVII. — Quaternion Investigation of the Commutative Law for 
Homogeneous Strains. By Frank L. Hitchcock. Communicated 
hy Dr C. G. Knott, General Secretary. 
(MS. received March 15, 1915. Eead June 7, 1915.) 
1. If a plastic body be subjected to a uniform change of shape, any 
two equal, similarly placed cubes become equal, similarly placed parallel- 
epipeds. It is well known that the character of the deformation may be 
determined by a set of nine constants ; but the physicist naturally prefers 
to regard the strain as an operator, and to represent it by a single symbol. 
The resulting operational algebra may even react favourably on the mathe- 
matical aspect of the question, and give us new methods of attack for old 
problems. 
The question. When are two homogeneous strains commutative in 
their order of application ? has never, so far as I am able to discover, been 
clearly answered. The problem depends, in fact, upon a more rigorous 
classification of various types of strain.* 
Homogeneous strains may be accurately divided into four classes, 
according as the number of distinct axes is three, or two, or one, or is 
infinite. There are no other possibilities : there must be at least one 
axis; j* and if there are four there are an infinite number. For if we have 
/3i, and /3g, vectors parallel to three diplanar axes, any fourth vector p 
may be expressed in terms of them. Suppose 
P ~ ^if^l •^2^2 ^3^3* • • • ’ ' ( ^ ) 
Let (p convert /5j, and /3g into ^ 2 /^ 2 ’ Then 
92^2^2'^ .... ( 2 )' 
But if p be a fourth axis, (pp = g^p, or, by (1), 
(j>p — "t" g^x^fSo. .... (3) 
* A general law for commutation of matrices is given by H. Taber, Am. Acad. Proc.y 
1891, 26, 64-66, but the result is ill adapted to physical interpretation. Several elegant 
examples are given by Gibbs, in the language of dyadics {Scientific Papers, ii, p. 63), but 
the discussion is not completed so as to cover all cases. 
C. J. Joly, in his Manual of Quaternions, gives the rule : Two linear vector functions 
are commutative when, and only when, they have the same axes. This rule holds whenever 
both strains have three distinct axes, but is otherwise insufficient, as exemplified in the text, 
t Kelland and Tait, Pntroduction to Quaternions, chap. x. 
