A MATHEMATICAL THEORY OF ELASTICITY. 
483 
The forces actually experienced by the sides A, B, C have nothing to balance them 
except the force actually experienced by K. Hence those three forces must have a 
single resultant, and the force on K must be equal and opposite to it. If, therefore, 
the force on K per unit of surface be denoted by F, and its direction cosines by 
I, m, n, we liave 
F.K.^ =P(X)A+T(Z)B+T(Y)C, 
F.K.w.=T(Z)A+Q(Y)B+T(X)C, 
F.K.w =T(Y)A+T(X)B+R(Z)C ; 
and, by the relations between the cases stated above, we deduce 
F/ =P(X) cos a+T (Z) cos/3+T(Y) cos 7, 
Fm=T(Z) cosa+Q(Y) cos/3+T(X) cos 7, 
F« =T(Y) cos a+T(X) cos /3+B(Z) cos 7. 
Hence the problem of finding (a, j 3 , 7), so that the force F(/, w, n) may be perpen- 
dicular to it, will be solved by substituting cos a, cos j 3 , cos 7 for /, m, n in these equa- 
tions. By the elimination of cos a, cos ( 3 , cos 7 from the three equations thus obtained, 
we have the well-known cubic determinantal equation, of which the roots, necessarily 
real, lead, when no two of them are equal, to one and only one system of three 
rectangular axes having the stated property. 
Def. The three lines thus proved to exist for every possible homogeneous stress 
are called its axes. The planes of their pairs are called its normal planes : the mutual 
forces between parts of the body separated by these planes, or the forces on portions 
of the bounding surface parallel to them, are called the principal tensions. 
Cor. 1. The Principal Tensions of the stress are the roots of the determinant cubic 
referred to in the demonstration. 
Cor. 2. If a stress be specified by the notation P(X), &c. as explained above, its 
normal planes are the principal planes of the surface of the second degree whose 
equation is 
P(X)X^-i-Q(Y)Y^H-R(Z)Z^-l-2T(X)YZ+2T(Y)ZX-f2T(Z)XY=l ; 
and its Principal Tensions are equal to the reciprocals of the squares of the lengths 
of the semi -principal-axes of the same surface (quantities which are negative of course 
for the principal axis or axes which do not cut the surface when the surface is a 
hyperboloid of one or of two sheets). 
Cor. 3. The ellipsoid whose equation referred to the Rectangular axes of a stress, is 
( 1 ~2eF)X^-}- (1 -2eG) Y^+ ( 1 - 2eH)Z^= 1 , 
where F, G, H denote the Principal Tensions, and e any infinitely small quantity, 
represents the stress, in the following manner: — 
From any point P in the surface of the ellipsoid draw a line in the tangent plane 
3 s 
MDCCCLVI. 
