RECENT ADVANCES IN SCIENCE 5 



rational. Some interesting applications to elliptic function 

 theory are given by the authors. 



D. M. Y. Somerville (part 5, p. 385) treats the singularities 

 of the algebraic trochoids. He determines the number of fun- 

 damental singularities for the various types of curves, and also 

 analyses the compound singularities at infinity. 



E. K. Wakeford (part 6, p. 403) writes on canonical forms. 

 The paper is posthumous, and formed the second part of a 

 dissertation of the author. Being almost complete in itself, 

 the Society has published it separately. The writer's object 

 was to establish the possibility of reduction to canonical form 

 rather than to find the reducing process. 



In the American Journal of Mathematics, xlii, i, p. 11, 

 H. D. Frary publishes an interesting note on the Green's func- 

 tion for a plane contour. Its object is rather the practical 

 than the exact solution of this problem, and a method is de- 

 veloped which should be of considerable service to the physicist 

 in special cases. The Green's function depends on two points 

 and on assigned conditions at a boundary of specified form, 

 while satisfying a differential equation — usually, as in this 

 instance, that of Laplace. The function G occurring most 

 frequently in the applications is zero over the boundary, and 

 if this can be found, Dirichlet's problem is soluble for any other 

 condition of the boundary within wide limits. For \iw =v {6) 

 on the boundary the solution is 



r ^^^ 



2'Trw = / vj--ds 



The ordinary methods of finding this function are of very 

 limited application. There is the method of images, which can 

 deal with the circle, semicircle, infinite strip, half plane, and 

 certain triangles, but for other contours, hyperelliptic integrals 

 are usually involved. The Schwarzian method, by the con- 

 formal representation of a polygonal boundary on a unit circle, 

 has not been extended very effectively beyond the point at 

 which Schwarz left it, for it again leads soon to abelian and 

 hyperelHptic integrals, though for regular polygons the solution 

 is capable of arithmetical evaluation and can be made practical. 



The author follows a new method, with obvious relations 

 to that of Fourier and Neumann. An infinite set of functions, 

 linearly independent, satisfying the differential equation sever- 

 ally, is selected, and the Green's function expanded in a series 

 of them with coefficients determined by the boundary conditions. 

 To find these coefficients, we have an infinite set of equations 

 in an infinity of unknowns, and the solution of these equations 

 is possible in a variety of cases. 



