May 26, 1911] 



SCIENCE 



807 



of variations would no longer present spe- 

 cial difficulty. 



To establish the point remaining to be 

 proved, Hilbert has used two different arti- 

 fices; he has not developed the first as com- 

 pletely as would be desirable, and has at- 

 tached himself especially to the second. 

 This consists in replacing the proposed 

 function u by the function v, which comes 

 from it by a double quadrature and of 

 which it is the second derivative with re- 

 gard to two independent variables. The 

 derivatives of v being the first integrals of 

 u, we can assign them an upper limit, by 

 the help of certain inequalities easy to 

 prove. Only it is necessary to be resigned 

 to a new circuit and to an artifice simple 

 however to apply to this new unknown 

 function v the rules of the calculus of vari- 

 ations which apply so naturally to the 

 function u. 



It is needless to insist upon the range of 

 these discoveries which go so far beyond 

 the special problem of Dirichlet. It is 

 not surprising that numerous investigators 

 have entered the way opened by Hilbert. 

 We must cite Levi, Zaremba and Fubini; 

 but I think we should signalize before all 

 Eitz, who, breaking away a little from the 

 common route, has created a method of 

 numeric calculus applicable to all the prob- 

 lems of mathematical physics, but who in it 

 has utilized many of the ingenious pro- 

 cedures created by his master Hilbert. 



Eecently Hilbert has applied his method 

 to the question of conformal representa- 

 tion. I shall not analyze this memoir in 

 detail. I shall confine myself to saying 

 that it supplies the means of making this 

 representation for a domain limited by an 

 infinite number of curves or for a simply 

 connected Riemann surface of an infinity 

 of sheets. This therefore is a new solution 

 of the problem of the uniformization of 

 analytic functions. 



DIVERS 



We have passed in review the principal 

 research subjects where Hilbert has left his 

 trace, those for which he shows a sort of 

 predilection and whither he has repeatedly 

 returned ; we must mention still other prob- 

 lems with which he has occupied himself 

 occasionally and without insistence. I 

 think I should confine myself to giving in 

 chronologic order the most striking results 

 he has obtained of this sort. 



Excepting the binary forms, the quad- 

 ratic forms and the biquadratic ternary 

 forms, the definite form most general of its 

 degree can not be broken up into a sum of 

 a finite number of squares of other forms. 



By elementary procedures may be found 

 the solutions in integers of a diophantine 

 equation of genus null. 



If an integral polynomial depending 

 upon several variables and several param- 

 eters is irreducible when these parameters 

 remain arbitrary, we may always give these 

 parameters integral values such that the 

 polynomial remains irreducible. 



Consequently there always exist equa- 

 tions of order n with integral coefficients 

 and admitting a given group. 



The fundamental theorem of Dedekind 

 about complex numbers with commutative 

 multiplication may be easily proved by 

 means of one of the fundamental lemmas 

 of Hubert's theory of invariants. 



The diophantine equation obtained by 

 equating to ± 1 the discriminant of an 

 algebraic equation of degree n has always 

 rational solutions, but save for the second 

 and the third degrees has no integral solu- 

 tions. 



Among the real surfaces of the fourth 

 order, certain forms logically conceivable 

 are not possible ; for example, there can not 

 be any composed of twelve closed surfaces 

 simply connected or of a single surface 

 with eleven perforations. 



