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PROFESSOR THOMSON’S ELEMENTS OF 
system of types of reference may also be called an orthogonal system. The elements 
specifying, with reference to such a system, any stress or strain, will be called ortho- 
gonal components or orthogonal coordinates. 
Examples. — (1) The six types described in Example (2) of Article VIII. are clearly orthogonal, if 
the parallelepiped referred to is rectangular. Three of these are simple longitudinal extensions, 
parallel to the three sets of rectangular edges of the parallelepiped. The remaining three are plane 
distortions parallel to the faces, their axes bisecting the angles between the edges. They constitute 
the system of types of reference uniformly used hitherto by writers on the theory of elasticity. 
(2) The six strains in which a spherical portion of the solid is changed into ellipsoids having the 
following equations — (1 -H A)X^-1- Y2 + Z^= 1 
X2 + (1 + B)Y2 + Z2=1 
. X2 + Y2 + (1 + C)Z2=1 
X2 + Y2 + Z2 + DYZ=1 
X2 + Y2 + Z2 + EZX = 1 
X2 + Y2 + Z2 + FXY=1, 
are of the same kind as those considered in the preceding example, and therefore constitute a nor- 
mal system of types of reference. The resultant of the strains specified, according to those equa- 
tions, by the elements A, B, C, D, E, F, is a strain in which the sphere becomes an ellipsoid whose 
equation (see above, Art. VII. Ex. (5)) is 
( 1 + A)X2 -h ( l-f B) Y2 -h ( 1 -F C)Z2 + D YZ + EZX + FXY = 1 . 
(.3) * A uniform cubical compression (L), three simple distortions having their planes at right 
angles to one another and their axesf bisecting the angles between the lines of intersection of these 
planes (11.) (III.) (IV.), any simple or compound distortion consisting of a combination of longi- 
tudinal strains parallel to those lines of intersection (V.), and the distortion (VI.), constituted from 
the same elements which is orthogonal to the last, afford a system of six mutually orthogonal types 
which will be used as types of reference below in expressing the elasticity of cubically isotropic solids. 
Article X. — On the Measurement of Strains and Stresses. 
Def. Strains of any types are said to be to one another in the same ratios as 
stresses of the same types respectively, when any particular plane of the solid 
acquires relatively to another plane parallel to it, motions in virtue of those strains 
which are to one another in the same ratios as the normal components of the forces 
between the parts of the solid on the two sides of either plane due to the respective 
stresses. 
Def. T he magnitude of a stress and of a strain of the same type, are quantities 
which, multiplied one by the other, give the work done on unity of volume of a body 
acted on by the stress while acquiring the strain. 
This example, as well as (7) of Art. X., (5) of XL, and the example of Art. XII., have been inserted to 
prepare for an application of the theory of Principal Elasticities to cubically and spherically isotropic bodies, 
added to the Second Part of this paper since the date of its communication. 
t The “ axes of a simple distortion” are the lines of its two component longitudinal strains. 
