THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 213 
By the same method as in the case of W, but using the theorem (i) of $40 
instead of Fourier’s heorem, we see that functions /f, (x,y) exist, solutions of 
(v?+«) f=0 and satisfying the above boundary equations. 
Thus the apportionment proposed for the edge tractions does actually satisfy the 
conditions to a first approximation. The solution found gives tractions of which 
the principal parts are the tractions actually assigned in the problem. The residual 
traction given by the solution is of the first and higher orders; and a second 
approximation to the problem will be obtained by subtracting a solution giving the 
residual tractions of the first order, such solution being found by the method used 
in the first approximation. 
This process would be tedious, and the way would be blocked at an ‘early stage 
by our ignorance of the coeflicients of the expansion (i) of § 40. 
We therefore pass at once to the consideration of the powerful method furnished 
by Bettis Theorem for the determination of the permanent mode. 
43. Hatensional strain. The Green's Function method for the permanent mode. 
The method has already been explained (§ 36). If we wish the permanent displace- 
ment at (x’, y’, 2’) in any direction (say the displacement ~), we take the permanent 
part of the solution for a unit force in that direction (a unit X force), modify it by 
removing the terms which convey resultant stress, and then try to balance it at the 
edge by adding a solution, without internal singularity, which shall neutralise its edge 
tractions. 
The displacements at the edge in the balanced solution, 7.e. in the solution obtained 
as the sum of the source and balancing solutions, being w’, v’, w’, or p’, q’, w’, and the 
given tractions X,Y, Z, or N,S, Z, we have 
ue, y', 2’) = / | (Xu! + Yo! + Zw'\ds de 
= | fox’ + Sq’ + Zw')ds dz 
the integral being taken over the cylindrical boundary. 
The thickness 2h being supposed infinitesimal, the object of the method is to deter- 
mine a few of the terms of p’, q’, w’ of lowest order in h. 
An alternative method would be to determine the functions E,, E, of § 33, I. (i), 
in terms of edge tractions; but the least confusing method of all is perhaps to 
determine 
dE, dE, Cie We, 
ie = Paya peat : 
iii, wh a tr 
These do not contain z’, but when they are known the complete solution can 
obviously be written down. We begin with U’, and in fact it will not be necessary to 
determine V’ separately. 
