492 
PROFESSOR THOMSON’S ELEMENTS OF 
Cor. 5. Ifa’, j/, z, are orthogonal components of any strain or stress, r, its 
concurrences with the types of reference are respectively 
X y z ^ ^ 
— ? — ? -? —5 — J — ? 
Z' Z' Z' y* 
where 
r = ( +3/^ -|- ^ . 
Cor. 6. The mutual concurrence of two stresses or strains is 
mm' -]- nn! 4“ 4- H" 
if (/, m, 71, X, fjb, v) denote the concurrences of one of them with six orthogonal types 
of reference, and (/', m', w', V, (jJ, v') those of the other. 
Cor. 7- The most convenient 6:joeci^cof/ow q/” « ^ 3 /joe for strains or stresses, being 
in general a statement of the components, according to the types of reference, of a 
unit strain or stress of the type to be specified, becomes a statement of its concurrences 
with the types of reference when these are orthogonal. 
Examples. — (1) The mutual concurrence of two simple longitudinal strains or stresses, inclined to 
one another at an angle S, is cos® d. 
(2) The mutual concurrence of two simple distortions in the same plane, whose axes are inclined 
at an angle $ to one another, is cos® fl — sin®5, or 2 sin (45° — fl) cos (45° — S). 
Hence the components of a simple distortion, 8, along two rectangular axes in its plane, and two 
others bisecting the angle between these taken as axes of component simple distortions, are 
8 (cos® $ — sin® fl) and 8 . 2 sin 5 cos 9 
respectively, if 9 be the angle between the axis of elongation in the given distortion and in the first 
component type. 
(3) The mutual concurrence of a simple longitudinal strain and a simple distortion is 
V 2. cos « cos /3, 
if « and /3 be the angles at which the direction of the longitudinal strain is inclined to the line 
bisecting the angles between the axes of the distortion ; it is also equal to 
X 
^ (cos®<p- cos® 4/), 
if <p and 4/ denote the angles at which the direction of the longitudinal strain is inclined to the axis 
of the distortion. 
(4) The mutual concurrence of a simple longitudinal strain and of a uniform dilatation is 
\ O 
(5) The specifying elements exhibited in Example (7) of the preceding article, are the concur- 
rences of the new system of orthogonal types described in Example (3) of Art. IX., with the ordi- 
nary system, Examples (1) and (2), Art. IX. 
Article XII. — 0)vthe Transformation of Types of Reference for Stresses or Strains. 
To transform the specification {x, y, z, f, ??, tf) of a stress or strain with reference 
to one system of types into ( <3? j j *^25 *^35 *^55 ) with reference to another system of 
types. Let (a„ />,, c,, ey,f, gj be the components, according to the original system, 
