1184 Proceedings of Royal Society of Edinburgh. [sess. 
Analytical Theory of the Equilibrium of an Isotropic Elastic 
Plate,” published in the Transactions of the Society. 
The problem of the deformation of an isotropic elastic plate 
under given forces has occupied the attention of mathematicians 
from the time of Lame. The solution given by Lame himself is 
merely formal ; the integrals by which that solution is expressed 
are not only very complicated, hut are not convergent, and they do 
not lead to the approximate theory. 
In his memoir Mr Dougall makes a new departure and 
develops a method that has important applications in other 
branches of applied mathematics. By an exceedingly skilful use 
of Cauchy’s theory of contour integration certain integrals, which 
in Lame’s solution are not convergent, are transformed into highly 
convergent series, and the modifications which are necessary to 
secure convergence lead at once to the most significant terms of 
the solution. The theorem of Betti is applied 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 that for an infinite plate, 
and here as elsewhere throughout the memoir numerous results 
are obtained which have great value both for pure and for applied 
mathematics. The memoir confirms the ordinary approximate 
theory, but extends it in various directions ; for example, the edge 
conditions given by Kirchhoff in correction of Poisson are found 
directly from the mathematical investigation, without the aid of 
any special physical hypothesis, and are carried to a higher degree 
of approximation than by Kirchhoff himself. The memoir 
contains much acute analysis, and strikes out a new method of 
treating the problems of mathematical physics that seems likely to 
be of great value in future investigations. 
The Chairman, on presenting the Neill Prize for 1901-1904 to 
Professor Graham Kerr, said : — 
The Neill Prize for the period 1901-1904 has been awarded to 
Professor John Graham Kerr, M.A., for his Researches on 
Lepidosiren paradoxa, published in the Philosophical Transactions 
of the Royal Society, London. 
This work includes an account of the embryological material 
