490 
PROFESSOR THOMSON’S ELEMENTS OF 
(3) A uniform dilatation in all directions, in which lineal dimensions are augmented in the ratio 
1:1+^’, is a strain equal in magnitude to ^v'3; or a uniform “cubic expansion” E is a strain 
, E 
equal to 
(4) A stress compounded of a uniform unit pressure in one direction and an equal tension in a 
direction at right angles to it, or which is the same thing, a stress compounded of two balancing 
couples of unit tangential pressures in planes at angles of 45° to the direction of those forces and at 
right angles to one another, amounts in magnitude to 2. 
(5) A strain compounded of a simple longitudinal extension, x, and a simple longitudinal con- 
densation of equal absolute value, in a direction perpendicular to it, is a strain of magnitude x\/2’, 
or, which is the same thing, (if — a simple distortion such that the relative motion of two planes 
at unit distances parallel to either of the planes bisecting the angles between the two planes men- 
tioned above is a motion tr, parallel to themselves, is a strain amounting in magnitude to 
(6) If a strain be such that a sphere of unit radius in the body becomes an ellipsoid whose equa- 
tion is 
( 1 - A)X2 + (1 - B) Y2 + (1 - C)Z2- D YZ - EZX - FXY = 1, 
the values of the component strains corresponding, as explained in Example (2) of Art. IX., to the 
different coefficients respectively, are 
iA, IB, W, 
D E F 
2^2’ 2V'2’ 2v'2' 
For the components corresponding to A, B, C are simple longitudinal strains, in which diameters 
of the sphere along the axes of coordinates become elongated from 2, to 2 + A, 2 -f B, 2 -f C respect- 
ively : D is a distortion in which diameters in the plane YOZ, bisecting the angles YOZ and Y'OZ, 
become I'espectively elongated and contracted from 2 to 2-j-|D, and from 2 to 2 — AD; and so for 
the others. Hence, if we take x, y, z, 0, rj, ^ to denote the magnitudes of six component strains, 
according to the orthogonal system of types described in Examples (1) and (2) of Art. IX., the 
resultant strain equivalent to them will be one in which a sphere of radius 1 in the solid becomes an 
ellipsoid whose equation is 
(l-2x)X^+{l- 2y) Y2 + (1 - 2z)Z^-2 ^2(0YZ + riZX + ?XY) = 1, 
and its magnitude will be 
V {x'^ + y"^ + z'^ + + Yf‘ + 
(7) The specifications, according to the system of reference used in the preceding Example, of 
unit strains belonging to the six orthogonal types defined in Example (3) of Art. IX., are respectively 
as follows : — 
X 
y 
z 
■n 
r 
(I-) 
1 
Vs 
1 
Vs 
1 
VS 
0 
0 
0 
(II.) 
0 
0 
0 
1 
0 
0 
(III.) 
0 
0 
0 
0 
1 
0 
(IV.) 
0 
0 
0 
0 
0 
1 
(V.) 
1 
m 
n 
0 
0 
0 
(VI.) 
1 
V 
wl 
n' 
0 
0 
0 
