July 25, 1902.] 



SCIENCE. 



127 



nitely thin plane screen bounded by a 

 straight edge or perforated with a circular 

 aperture. ' ' 



Again the same author says in his Avork 

 on the ' Theory of Sound ' :* 



"The diftraction of sound is a subject 

 which has attracted but little attention 

 either from mathematicians or experiment- 

 alists. Although the general character of 

 the phenomena is well understood, and 

 therefore no very striking discoveries are 

 to be expected, the exact theoretical solu- 

 tion of a few of the simpler i^roblems, 

 which the subject presents, would be in- 

 teresting. ' ' 



Accordingly the recent solutions of Som- 

 merfeld and Carslaw are very welcome to 

 mathematicians and physicists. A very 

 brief sketch of the principle of the method 

 may here be given. 



Let the waves come from the direction 

 ^ = 0', and be incident on the plane 6* = 0. 

 In the {x, y) plane, or in the {r,()) plane, 

 the origin will be regarded as a winding 

 point, and the line o = x -\- O' a branch 

 line. Start with the simplest solution of 

 our differential equation, namely, that for 

 Tindisturbed plane waves in infinite space, 



replace 0' by «, multiply by the same two- 

 valued function of a as before, and inte- 

 grate around the point«=<^'in the complex 

 «-plane. The result of the integration is a 

 multiform solution of period 47r. The 

 solution of the physical problem is obtained 

 by adding the multiform solution for Avaves 

 ■coming from the direction 0' to that for 

 the direction —6'. There is, of course, 

 considerable difficulty in performing the in- 

 dicated operations, but this does not dimin- 

 ish the theoretical value of the soliition, as 

 the difficulties belong only to the integral 

 calculus. 



* ' Theory of Sound,' Vol. II., p. 141. 



The next problem in order is that of 

 waves issuing from a point-source against 

 the half-plane, either in two or in three di- 

 mensions. 



In the latter case we start with the undis- 

 turbed solution in infinite space 



and treat this function as we treated 1/R 

 in the potential problem. We put poles at 

 (r', 0', o) and (r', — 0', o), and take the 

 physical space as defined by < ^ < 27r. 

 It will be found that the function 



satisfies all the conditions in the assigned 

 physical space. 



In the corresi^onding two-dimensional 

 problem, the starting point is the undisturb- 

 ed solution 



u= Toiler), 



where Y^ is the Nemnann function. 



The same method is applicable to prob- 

 lems in the flow of heat, ii;i which the equa- 

 tion 



is to be satisfied. The starting point is the 

 solution for a point-source in an infinite 

 solid 



_ 1 



"-(4-H)? 





du 



' \ ikt /■ 



In his recent paper published in the Zeit- 

 schrift, Sommerfeld has extended the meth- 

 od so as to apply to the problem of Rontgen 

 rays encountering an obstacle represented 

 by the same half-plane. He obtains a 

 multiform solution of Maxwell's equations, 

 and adapts it to the physical conditions, 

 comparing the results with experimental 

 data. 



The induced currents flowing in an infi- 

 nite half plane have been studied by Mr. 

 Jeans* by the multiform method, using a 



*Proc. Lond. Math. Soc, 1899. 



