126 



SCIENCE. 



[N. S. Vol. XVI. No. 395. 



Avhere u is of the same form as u-^ written 

 above. 



By inversion mtli regard to different 

 centers, various other problems are reduced 

 to this one; for instance, the infinite plane 

 with a circular aperture, the circular disc, 

 .and the spherical segment. 



With regard to the uniqueness of the 

 ■solution, Dr. Sommerfeld has proved by a 

 remarkable use of function-theory methods 

 that a function satisfying the conditions 

 already laid down for Green's function is 

 uniquely determined in a Riemann space. 



I next speak of some recent advances in 

 the solution of an equation more general 

 than Laplace's, namely, the differential 

 equation 



d'u j^ dht 



khi = 0, 



which plays such an important part in the 

 treatment of vibrating systems of various 

 kinds ; and I may introduce them hj a quo- 

 tation from Pockels' treatise on this equa- 

 tion : ' ' Those solutions of our differential 

 equation, which in accordance with their 

 physical significance are regarded as single- 

 valued within certain bounded regions, 

 Avould by analytical continuation over 

 the boundary in general become multiform. 

 Therefore, from both a mathematical and 

 a physical standpoint, multiform functions 

 are important, and it is very desirable that 

 the properties of such functions, their 

 winding points and singularities, their be- 

 havior on Riemann surfaces, etc., should 

 be systematically investigated — in short, 

 all the function-theory questions which 

 were handled in the theory of the New- 

 tonian and logarithmic potential. * * * 

 ' ' Similarly as we have treated of solu- 

 tions that are single- valued in the whole 

 plane, it would be of interest to seek func- 

 tions which are single-valued on a closed 

 Riemann surface, or in an analogous three 

 dimensional region, more especially those 



functions which are everywhere iinite and 

 continuous, namely the so-called ' principal 

 solutions,' within the region in question. 

 Finally there is the further investigation of 

 the essential singularities and the natural 

 boundaries wliich the functions satisfying 

 this equation may present. * * * Inves- 

 tigations regarding these questions" have 

 not yet been made, more especially the in- 

 tegration of our equation for a closed mani- 

 fold has hardly been touched. In this di« 

 rection of inquiry without doubt a wide 

 and rich field offers itself." 



These words were written in 1890 ; and 

 in 1897 appeared Professor Sommerfeld 's 

 paper on multifojrm potentials of which I 

 have given some account above. He and 

 his pupil. Dr. Carslaw, have also attacked 

 the multiform solutions of the more general 

 equation to which Pockels refers.* 



The first problem that presents itself is 

 to find a solution that has no pole, and is 

 multiform with period 2nT, in the ordinary 

 sense, biit on a certain ?i-sheeted Riemann 

 surface is uniform. The case ■)i=2 solves 

 the following well-lcnown physical prob- 

 lem: 



Plane waves of sound, light or electricity 

 are incident on a thin infinite half plane 

 bounded by a straight edge, to find the re- 

 sulting diffraction of the waves. 



This problem had previously been men- 

 tioned by Lord Rayleigh in the article on 

 Wave Theory in the Encyclopajdia Britan- 

 nica in the following terms : 



' ' The full solution of problems concern- 

 ing the mode of action of a screen is scarce- 

 ly to be expected. Even in the simple case 

 of sound where we know what we have to 

 deal with the mathematical difficulties are 

 formidable, and we are not able to solve 

 such an apparently elementary question as 

 the transmission of sound past a rigid infi- 



* Proc. Land. Math. Soc, 1898; Zeitschrift, 

 1901; Proc. Edin. Math. Soc, 1901. 



