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PROFESSOR THOMSON’S ELEMENTS OF 
half-way to the point where this plane is cut by a perpendicular to it through the 
centre ; and from the end of the first-mentioned line draw a radial line to meet the 
surface of a sphere of unit radius concentric with the ellipsoid. The tension at this 
point of the surface of a sphere of the solid is in the line from it to the point P; and 
its amount per unit of surface is equal to the length of that infinitely small line, 
divided by e. 
Article IV. — On the Distrihution of Displacement in a Strain. 
Prop. In every homogeneous strain any part of the solid bounded by an ellipsoid, 
remains bounded by an ellipsoid. 
For all particles of the solid in a plane remain in a plane, and two parallel planes 
remain parallel. Consequently every system of conjugate diametral planes of an 
ellipsoid of the solid retain the property of conjugate diametral planes with reference 
to the altered curve surface containing the same particles. This altered surface is 
therefore an ellipsoid. 
Prop. There is a single system (and only a single system, except in the cases of 
symmetry) of three rectangular planes for every homogeneous strain, which remain 
at right angles to one another in the altered solid. 
Def. These three planes are called the normal planes of the strain, or simply the 
strain-normals. Their lines of intersection are called the axes of the strain. 
Remark. The preceding propositions and definitions are applicable, to whatever 
extent the body may be strained. 
Prop. If a body, while experiencing an infinitely small strain, be held with one 
point fixed and the normal planes of the strain parallel to three fixed rectangular 
planes through the point, O ; a sphere of the solid of unit radius having this point 
for its centre becomes, when strained, an ellipsoid whose equation, referred to the 
strain-normals through O, is 
( 1 — 2x)X" -f- ( 1 — 23/) Y" + ( 1 - 22 )Z" = 1 , 
\(x,y,z denote the elongations of the solid per unit of length, in the directions 
respectively perpendicular to these three planes ; and the position, on the surface of 
this ellipsoid, attained by any particular point of the solid, is such that if a line be 
drawn in the tangent plane, half-way to the point of intersection of this plane with a 
perpendicular from the centre, a radial line drawn through its extremity cuts the 
primitive spherical surface in the primitive position of that point. 
Cor. For every stress, there is a certain infinitely small strain, and conversely, for 
every infinitely small strain, there is a certain stress, so related that if, while the 
strain is being acquired, the centre and the strain-normals through it are held fixed, 
the absolute displacements of particles belonging to a spherical surface of the solid 
represent, in intensity (according to a definite convention as to units for the repre- 
sentation ol force by lines), and in direction, the force (reckoned as to intensity, in 
