55 
of Edinburgh, Session 1869 - 70 . 
If a plane sheet is in equilibrium under the action of internal stress 
of any kind, then a quantity, which we shall call Airy’s Function 
of Stress, can always be found, which has the following properties. 
At each point of the sheet let a perpendicular be erected pro- 
portional to the function of stress at that point, so that the 
extremities of such perpendiculars lie in a certain surface, which 
we may call the surface of stress. In the case of a plane frame the 
surface of stress is a plane-faced polyhedron, of which the frame is 
the projection. On another plane, parallel to the sheet, let a per- 
pendicular be erected of height unity, and from the extremity of 
this perpendicular let a line be drawn normal to the tangent 
plane at a point of the surface of stress, and meeting the plane at 
a certain point. 
Thus, if points be taken in the plane sheet, corresponding points 
may be found by this process in the other plane, and if both points 
are supposed to move, two corresponding lines will be drawn, which 
have the following property: — The resultant of the whole stress 
exerted by the part of the sheet on the right hand side of the line 
on the left hand side, is represented in direction and magnitude 
by the line joining the extremities of the corresponding line in 
the other figure. In the case of a plane frame, the corresponding 
figure is the reciprocal diagram described above. 
From this property the whole theory of the distribution of stress 
in equilibrium in two dimensions may be deduced. 
In the most general case of three dimensions, we must use three 
such functions, and the method becomes cumbrous. I have, however, 
used these functions in forming equations of equilibrium of elastic 
solids, in which the stresses are considered as the quantities to be 
determined, instead of the displacements, as in the ordinary form. 
These equations are especially useful in the cases in which we 
wish to determine the stresses in uniform beams. The distribution 
of stress in such cases is determined, as in all other cases, by the 
elastic yielding of the material ; but if this yielding is small and 
the beam uniform, the stress at any point will be the same, what- 
ever be the actual value of the elasticity of the substance. 
Hence the coefficients of elasticity disappear from the ultimate 
values of the stresses. 
In this way I have obtained values for the stresses in a beam 
