156 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
In (33) and (34) each line represents a potential function; in (83) the first lines | 
define a particular solution giving the proper values of the tractions at the surface, as 
may be seen from (13); the partial solutions given by the « series give zero surface — 
tractions, and represent a strain insensible except in the vicinity of p=a@; and the 
solutions defined by the last lines, being of the form (23), give zero surface tractions. 
From these remarks it follows immediately that the solution (33), (34) satisfies all 
the conditions of the problem in the two regions p<a,p>a, taken separately. To 
verify the solution completely, it would be necessary to show in addition that certain 
conditions are fulfilled at the surface p=a, namely, 
(i) that the displacements and strains are continuous at this surface, and 
(ii) that the integral value of the stresses za, zy, 22 over any small area lying partly 
within and partly without the cylinder p=qa, on either of the plane faces of the plate, 
tends to zero when the area is indefinitely diminished. 
The condition (i) ensures the ‘ synexis’ of the solution across the surface p=a, and 
can be proved by showing, as may easily be done by means of summations similar to — 
those of § 11, that ¢, 6, at and ug are continuous at that surface. For by the Theory — 
Ip p 
d 
of the Potential this carries with it the continuity of all the derivatives of p and @, and 
therefore of the displacements and stresses, as well as of all their derivatives, under the — 
proviso, of course, that-—h<z<h. 
The condition (ii), or some equivalent, is required in order to exclude the possibility 
of stresses with finite resultant passing into the solid through the Imes z=+h, p=a; 
or, in other words, in order to ensure that the solution is not partly due to linear 
~ 
elements of traction at these lines. 
18. Order of the various parts of the solution, when h is small. 
The final form of the solution, as exhibited in (33), (34) was obtained by combining ~ 
parts of ¢,, , with @,, 9, and until this was done, it was not immediately evident — 
that @ and @ were potentials. Thus the part of the solution arising from the imaginary 
values of «, or from any one of them, is not, within the region of applied traction, a 
potential by itself, and the same is true of the ¢,, 9, part, which may be considered as 
coming from the zero values of «. This has sometimes to be taken into account in © 
calculating the stresses ; the formula for 2, for example, in (5) requires additional terms 
if, while w, v, w are still given by (4), ¢ and 6 are not potentials. 
On the other hand, the separation of the solution into the two parts (29), (30) has 
this very marked advantage that, when h is very small, the first part gives the terms of 
the two lowest orders in h of $, 8, namely those of orders h~* and h°, while the second 
part, as we have already seen, contains no terms of lower order than /”. When, how- 
ever, we come to calculate displacements and stresses, the separation is less simple, 
mainly in consequence of the fact that «, y differentiations do not change the order of 
Pp; , 9, , but diminish the order of $,, 0, by one for each differentiation. 
