806 



SCIENCE 



[N. S. Vol. XXXIII. No. 856 



tions of a complex variable subject to cer- 

 tain conditions as to the limits, to the the- 

 orem of oscillations of Klein, to the forma- 

 tion of fuchsian functions, and in particu- 

 lar to the following problem: to determine 

 A so that the equation 



[^(,._a)(.,_5)(^_e)g] + (., + A)3, = 



may be a fuchsian equation. 



One of the most unexpected applications 

 is that Hilbert makes to the theory of the 

 volumes and surfaces of Minkowski, and 

 by which he connects with Fredholm's 

 method a question important for those who 

 interest themselves in the philosophic an- 

 alysis of the fundamental notions orf 

 geometry. 



dieichlet's principle 

 We know that Riemann with a stroke of 

 the pen proved the fundamental theorems 

 of Diriehlet's problem and conformal rep- 

 resentation, grounding himself on what he 

 called Diriehlet's principle; considering a 

 certain integral depending upon an arbi- 

 trary function JJ, and which we shall call 

 Diriehlet's integral, he showed that this 

 integral can not become null and from this 

 he concludes that it must have a minimum, 

 and that this minimum can be reached only 

 when the function U is harmonic. This 

 reasoning was faulty, as has since been 

 shown, because it is not certain that the 

 minimum can be actually reached, and if it 

 is, that it can be for a continuous function. 

 Yet the results were exact; much work 

 has been done on this question ; it has been 

 shown that Diriehlet's problem can always 

 be solved, and it actually has been solved ; 

 it is the same with a great number of other 

 problems of mathematical physics which 

 formerly would have seemed attackable by 

 Riemann 's method. Here is not the place 

 to give the long history of these researches ; 

 I shall confine myself to mentioning the 



final point of outcome, which is Fredholm 's 

 method. 



It seemed that this success had forever 

 cast into oblivion Riemann 's sketch and 

 Diriehlet's principle itself. Yet many re- 

 gretted this; they knew that thus we were 

 deprived of a powerful instrument and 

 they could not believe that the persuasive 

 force which in spite of all Riemann 's argu- 

 ment retained, and which seemed to rest 

 upon I know not what adaptation of 

 mathematical thought to physical reality, 

 was actually only a pure illusion due to 

 bad habits of mind. Hilbert wished to try 

 whether it would not be possible, with the 

 new resources of mathematical analysis, to 

 turn Riemann 's sketch into a rigorous 

 proof. See how he arrived at it; consider 

 the aggregate of functions U satisfying 

 proposed conditions; choose in this aggre- 

 gate an indefinite series of functions S, 

 such that the corresponding Dirichlet in- 

 tegrals tend in decreasing toward their 

 lower limit. It is not certain that at each 

 point of the domain considered this series 

 S is convergent; it might oscillate between 

 certain limits. But we can in 8 detach a 

 partial series S^ which is convergent at a 

 point Mj of the domain; in Sj, detach an- 

 other partial series S^ which shall always 

 be convergent at M^, but which, moreover, 

 shall also be convergent at M^- So con- 

 tinuing, we shall obtain a series which will 

 be convergent at as many points as we 

 wish ; and by a simple artifice we from this 

 deduce another series which will be con- 

 vergent at all the points of a countable 

 assemblage, for example at all the points 

 whose coordinates are rational. If then 

 we could prove that the derivatives of all 

 the functions of the series are less in abso- 

 lute value than a given limit, we might 

 conclude immediately that the series con- 

 verges uniformly in the whole domain and 

 the application of the rules of the calculus 



