276 
MR. MACQUORN RANKING ON AXES OF 
20. Of Stresses, as referred to Oblique Axes. 
It is next required to express the stress at any particle of an elastic solid by means 
of a system of six elementary stresses which shall be contragredient to the system of 
six elementary strains defined in the preceding article. This is accomplished in the 
following manner. 
It is known that the total stress at any point may be resolved into three normal 
stresses on the three principal planes of the tasimetric surface. Let the direction 
and sign of any one of those three 'principal stresses be represented by those of a line 
OR, and its magnitude, as reduced to unity of area of the plane normal to that 
direction, by the square of that line. 
Let u, V, w be the contraordinates of R, as referred to the oblique axes OX, OY, 
OZ. Then will the stresses on unity of area of planes normal to those axes, in the 
direction OR, be represented respectively by 
ur, vr, wr. 
Let the Elementary Stresses be defined to be, the projections on the three axes of 
coordinates, of the total stresses on unity of area of the three pairs of faces of a paral- 
lelopiped, normal to the three axes respectively'. — then, if we take S to denote the 
summation of three terms arising from the three principal stresses, the elementary 
stresses will be expressed as follows : — 
Normal Stresses on the faces normal to 
57, y, 
Pj=:S.M^; P2=S.i!;^; 
Oblique Stresses on the faces normal to 
y z z X 
in the directions y x z 
Q2 = S.WU; 
Ps=S.W^ ; 
> • 
( 41 .) 
X y 
y X 
Q3=S.^^^;. ^ 
These expressions fulfil the condition of making the elementary stresses 
PlJ I*2J P35 QlJ Q2J Q3 
contravariant respectively to the elementary strains 
f^, 7, P', V, 
so that for oblique axes, as for rectangular axes, the potential energy of elasticity is 
represented by 
U= — 5(PiaH-P2/3+P37-j-Qi?.-l-Q2jM<+Q3j'), 
the universal concomitant ; and may be expressed either by a homogeneous quadratic 
function of the six elementary strains (as in equation 2), with twenty-one tasinomic 
coefficients, or by a homogeneous quadratic function of the six elementary stresses. 
