THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 165 
18. General solution. Comparison with the solution for normal traction. 
The solution of the general problem of (41) may be found by multiplying the 
expressions for ¥’, 6’, p’ in (48) . . . . (51) by sai («,y') and integrating term by term 
over the area A within which f is finite. As in the case of the problem of given 
normal traction, term by term differentiations of the resulting series will be legitimate 
| just so far as the derivatives are required for the calculation of displacements and 
stresses. In order to see this, it should be noticed that, while an extra differentiation 
as to x or y will be required in virtue of (43), the series for V’, 0’, p’ have general 
terms of one order higher in 1/« as compared with those of (20), (21). One effect, 
however, of this additional differentiation will be to increase the relative importance at 
the edge of the area A of that part of the displacement and stress which arises from 
the local perturbation, such displacement being. of order 4, and stress of order zero, as 
in the former problem, whereas the displacement and stress as a whole is of higher 
order in / than before. 
The functions ’, 6’, ¢’ being symmetrical about the axis R=0, it is clear that the 
solution for an element of traction of 87m units at («’, y’, 4) parallel to the axis of y is 
given by 
dy! do’ dg 
teeny a . ti 
with , 6’, ¢’ asin (48)... . (51). 
It will be seen presently that surface traction may be regarded as a special case of 
| force applied in the body of the plate. We may therefore postpone any more extended 
development of the above solution, and in particular any more explicit comparison of 
the results with those of the accepted approximate theory of thin plates, until we 
| have obtained the solutions of the problems relative to sources of strain situated in 
the interior of the solid. 
19. Normal force applied at a single internal point. Solution 
m definite mtegrals. 
“¢ take first the case of a single force, say for convenience of 47u4(a+1) units, 
applied at («’, y’, 2’) parallel to Oz, the faces of the plate being free from stress. 
Referring to (6), we see that the conditions of the problem may be taken to be 
(i) U2 @-) 40 | 
y= Cee : Ley (55) 
oe 
(ii) U, V, W, sae with their derivatives as to (x, y, z) of the Fc order, are finite 
and continuous at every point of the solid at a finite distance, and have derivatives of 
