130 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
analytical difficulties in the way of deducing them rigorously, they are either simply q 
assumed, or else supported by reasoning plausible rather than demonstrative. 
In the following pages the problem is treated as a purely mathematical one, and the 
approximate theory for a finite plate deduced from an exact solution for an infinite plate. 
The main features of the method are— 
(i) The use of Bessel functions in ‘place of the simple harmonic functions of previous 
writers. Only the symmetrical forms, or functions of order zero, are required. 
(ii) Transformation of the definite integrals, in terms of which the solutions are in 
the first place obtained, into series, by means of Cauchy’s theory of contour integration 
and residues. The series involve Bessel functions of the second kind with complex argu- 
ment, and are so highly convergent that the principal features of the strain represented 
by the solution can be made out with the utmost ease. (The transformations belong 
to a class discussed systematically, apparently for the first time, in a paper “On the 
Determination of Green’s Function by means of Cylindrical and Spherical Harmonies,” 
Proc. Edin. Math. Soc., vol. xviii.) 
(iii) Detailed solutions of the problems of internal force with vanishing face traction. 
The usual method of dealing with a general problem in Elasticity is to find a particular 
solution for the bodily force, and then to treat the problem of surface tractions 
completely. This is theoretically sufficient, but leaves the result in a complicated form, 
which in the present case must be simplified before practical applications can be made. 
(iv) Use of Betti’s Theorem (Love, Elasticity, vol. i. § 140) to develop a method 
analogous to the method of Green’s function in the Theory of the Potential, by which the 
properties of the solution for a finite plate can be deduced from the infinite plate solu- 
tion. (Cf. Proc. Edin. Math. Soc., vol. xvi., “On a general Method of Solving the 
Equations of Elasticity.”) 
The results of the ordinary theory are fully confirmed, and extended in various direc- 
tions. The infinite solid solution gives, of course, an exact particular solution for 
internal force and traction on the plane faces of a finite plate. At the head of the solu- 
tion appear the terms given by the approximate theory. In the case of flexure, the 
equation of Lagrange is obtained to a second approximation. 
The problem of a finite plate under given edge tractions cannot be completely solved, 
but exact solutions are given of certain problems relating to a circular plate. Fora thin 
plate, with edge of any shape, the conditions satisfied at the edge by the principal terms 
of the exact solution are found to a degree of approximation beyond the reach of any 
theory which rests merely on the “ principle of the elastic equivalence of statically equi- 
pollent loads.” For example, the celebrated boundary conditions given by KircHHoFF, in 
correction of Porsson, are verified, and extended by the inclusion of terms of higher order. 
In conclusion, it may be mentioned that the methods given here are equally appli- 
cable to the problem of the vibrations of a plate, and to the problems of the equilibrium 
and vibration of a finite circular cylinder, or of an open spherical shell. Some account 
of these applications I hope to publish shortly. 
