July 25, 1902.] 



SCIENCE. 



123 



In proving- that a solution always exists, 

 Dirielilet began by assuming as self-evident 

 that among all the functions which satisfy 

 the assigned boimdary conditions, there is 

 a certain function, u, for which the in- 



taken throughout the region T, is a mini- 

 mum. This assumption is usually called 

 ' Dirichlet 's principle. ' If this principle 

 be gTanted it can be shown by the calculus 

 of variations that the function u satisfies 

 Laplace 's equation ; and it is easy to prove 

 by Green's theorem that there is no other 

 solution. 



It was first pointed out by Weierstrass 

 that this assumption is not allowable. If 

 only a finite number of quantities present 

 themselves we can assume that there is a 

 smallest one among them. But among an 

 indefinite number of quantities in any as- 

 signed group a smallest one does not neces- 

 sarily exist. . Consider for instance those 

 rational niunbers which decrease towards 

 the square root of 2 as a limit ; there is no 

 smallest among them. 



This led mathematicians to seek for other 

 proofs of the existence theorem ; and many 

 interesting developments in function 

 theory have been the result. Very re- 

 cently Hilbert has reexamined Dirichlet 's 

 assumption, and has succeeded in demon- 

 strating it, so that it is once more available 

 as a starting point for the existence 

 .heorem. 



AVhen the boundary of the region is 

 rectangular, circular, spherical, cylindrical, 

 conical or ellipsoidal, the appropriate har- 

 monic functions will be found in such 

 woi'ks as Byerly's 'Fourier Series and 

 Spherical Harmonics.' 



I may mention here a new method of 

 obtaining solutions of Laplace's three- 

 dimensional equation used by Dr. Harris, 



and applied to tidal problems.* He uses 

 the more general complex variable contain- 

 ing two imaginary units i and j. An 

 arbitrary function of the form 



<l>{ax-\- Wy+jcz) 



is a solution of Laplace's equation, pro- 

 vided ^"^/^ — 1, and a' ^h'' -\- c' . When 

 this function is expanded, the real part, 

 and, the coefficients of i, of j and of ij, are 

 all separate solutions of the differential 

 equation. A great number of solutions of 

 this and similar equations can be obtained 

 by this method. It is to be hoped that Dr. 

 Harris may have time to develop it further. 



In order to lead up to some recent appli- 

 cations of function theory I wish to speak 

 especially of another method of solving 

 Dirichlet 's problem, namely by the use 

 of Green's function. 



Green's function is defined as follows 

 for a given closed boundary S and a given 

 pole Pj, within the bounded region T. 



Let {x, y, z) be the current point within 

 the region, and let (ajj, y^, sj be the pole. 

 Then G';^-! is to vanish at every point 

 of the bounclary S, and is to be harmonic 

 within the region T except at the pole 

 {_x^, 2/i, 2i), where it is to become infinite 

 as 1/r, where r is the distance of the cur- 

 rent point {x, y, z) from {x^, y^, z^). 



There is always one and only one Green's 

 function for a given boundary and pole. 

 The determination of the form of this func- 

 tion G furnishes a solution of Dirichlet 's 

 problem; for it has the property that the 

 surface integral 



//' 



taken over the boundary of S, has the 

 value i- Y{x^, y-^, sj, where V is any func- 

 tion harmonic within >S', and dG/dn is the 

 normal derivative of Green's function. 

 Hence the value of V at any point 

 * ' ManiiEil of Tides,' Part IV., A, pp. 584, 597. 



