486 
PROFESSOR THOMSON’S ELEMENTS OF 
(2) A simple extension or contraction in parallel lines unaccompanied by any transverse extension 
or contraction, that is, “ a simple longitudinal strain,” is orthogonal to any similar strain in lines at 
right angles to those parallels. 
(3) A simple longitudinal strain is orthogonal to a simple tangential strain*” in which the 
sliding is parallel to its direction or at right angles to it. 
(4) Two infinitely small simple tangential strains in the same plane fj with their directions of 
sliding mutually inclined at an angle of 45°, are orthogonal to one another. 
(5) An infinitely small simple tangential strain is orthogonal to every infinitely small simple tan- 
gential strain, in a plane either parallel to its plane of sliding or perpendicular to its line of sliding. 
Article VII. — Comjjosition and Resolution of Stresses and of Strains. 
Any number of simultaneously applied hoinog-eneous stresses are equivalent to a 
single homogeneous stress which is called their resultant. Any number of super- 
imposed hoinogeneous strains are equivalent to a single homogeneous resultant strain. 
Infinitely small strains may be independently superimposed ; and in what follows it 
will be uniformly understood that the strains spoken of are infinitely small, unless 
the contrary is stated. 
Examples. — (1) A strain consisting simply of elongation in one set of parallel lines, and a strain 
consisting of equal contraction in a direction at right angles to it, applied together, constitute a 
single strain, of the kind whieh that described in Example (3) of the preceding article is when infi- 
nitely small, and is called a plane distortion, or a simple distortion. It is also sometimes called a 
simple tangential strain, and when so considered, its plane of sliding may be regarded as either of 
the planes biseeting the angles between planes normal to the lines of the component longitudinal 
strains. 
(2) Any two simple distortions in one plane may be reduced to a single simple distortion in the 
same plane. 
(3) Two simple distortions not in the same plane have for their resultant a strain which is a 
distortion unaccompanied by change of volume, and which may be called a compound distortion. 
(4) Three equal longitudinal elongations or condensations in three directions at right angles to 
one another, are equivalent to a single dilatation or condensation equal in all directions. The single 
stress equivalent to three equal tensions or pressures in directions at right angles to one another is 
a negative or positive pressure equal in all directions. 
(5) If a certain stress or infinitely small strain be defined (Art. III. Cor. 3, or Art. IV.) by the 
ellipsoid 
(l+A)X2 + (i+B)YV(l+C)Z2 + DYZ + EZX-fFXY=]., 
and another stress or infinitely small strain by the ellipsoid 
( 1 + A')X2 -f ( 1 -p B') Y2 -f ( 14- C')Z2 -f- D' YZ + E'ZX -t- F'X Y = 1 , 
where A, B, C, D, E, F, &c. are all infinitely small, their resultant stress or strain is that repre- 
* That is, a homogeneous strain in which all the particles in one plane remain fixed, and other particles are 
only displaced parallel to this plane. 
[ “ The plane of a simple tangential strain,” or the plane of distortion in a simple tangential strain, is a 
plane perpendicular to that of the particles supposed to be held fixed, and parallel to the lines of displacement 
of the others. 
