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PROFESSOR. THOJMSON’S ELEMENTS OF 
unit of surface as that between the j3arts on the two sides of any parallel plane ; and 
the former tension or pressure is parallel to the latter. 
A strain is said to be homogeneous throughout a body, or the body is said to be 
homogeneously strained, when equal and similar portions with corresponding lines 
paralkd, experience equal and similar alterations of dimensions. 
Cor. All the particles of the body in parallel planes remain in parallel planes, 
when the body is homogeneously strained in any way. 
'Examples. x\ long uniform rod, if pulled out, Vv'ill experience a uniform strain, except near its ends. 
In a pillar bearing a weight in comparison with which its owm may be neglected, there will be a 
sensible heterogeneousness of the strain up to the middle from each end, because of the circum- 
stances that prevent the ends from expanding laterally to the same extent as the middle does. 
A piece of cloth held in a plane and distorted so that the warp and woof, instead of being perpen- 
dicular to one another, become two sets of parallels cutting one another obliquely, experiences a 
homogeneous strain. The strain is heterogeneous as to intensity, from the axis to the surface of a 
c}dindrical wire under torsion, and heterogeneous as to direction in different positions in a circle 
round the axis. 
Article OI. — On the Distribution of Force in a Stress. 
Theorem. In every homogeneous stress there is a system of three rectangular planes, 
each of which is perpendicular to the direction of the mutual force between the 
parts of the body on its two sides. 
For let P(X), P(Y), P(Z) denote the components, parallel to X, Y, Z, any three 
rectangular lines of reference, of the force experienced per unit of surface at any por- 
tion of the solid bounded by a plane parallel to (Y, Z) ; Q(X), Q(Y), Q(Z) the cor- 
responding components of the force experienced by any surface of the solid parallel 
to (Z, X) ; and R(X), R(Y), R(Z) those of the force at a surface parallel to (X, Y). 
Now by considering the equilibrium of a cube of the solid with faces parallel to the 
planes of reference, we see that the couple of forces Q(Z) on its two faces perpendi- 
cular to Y is balanced by the couple of forces R(Y) on the faces perpendicular to 
Z. Hence we must have 
Q(Z)=R(Y). 
Similarly, it is seen that 
R(X) = P(Z), 
and 
P(Y) =Q(X). 
For the sake of brevity, these pairs of equal quantities, being tangential forces 
respectively perpendicular to X, Y, Z, may be denoted by T(X), T(Y), T(Z). 
Consider a tetrahedral portion of the body (sunonnded it may be with continuous 
solid) contained within three planes A, B, C, through a point O parallel to the planes 
of the pairs of lines of reference, and a third plane K cutting these at angles a, /3, y 
respectively ; so that as regards the areas of the different sides we shall have 
A = Kcosc{, B = Kcos|3, C = Kcosy. 
