Decembee 26, 1913] 



SCIENCE 



903 



this work Poincare's contributions were 

 fundamental. His first article "Sur les 

 Equations aux Derivees Partielles de la 

 Physique Mathematique" appeared in the 

 American Journal of Mathematics in 1890. 

 The equations of mathematical physics are 

 all very similar and may be practically 

 all reduced to three or four. Of these the 

 equation of Laplace, 



SW SW S'V 



dx- dy^ Sz- ' 



is the most important. The so-called boun- 

 dary problem of finding a solution of La- 

 place's equation, valid for a certain region 

 of space, that shall take prescribed values 

 at the surface bounding this space is known 

 as Dirichlet's problem. Of this the prob- 

 lem of the distribution of electricity on the 

 surface of a conductor is a particular case, 

 the function given on the surface reducing 

 to a constant. 



The latter example is a case of the out- 

 side problem, where in addition we have 

 the condition that the desired function 

 must vanish at infinity. The demonstra- 

 tion of the existence of such a function 

 given by Riemann and depending upon the 

 application of the calculus of variations to 

 the definite integral 



///[(£)*+ (5)' +(f )■]-»- 



is lacking in rigor and the attempt to re- 

 place it has engaged the attention of some 

 of the greatest mathematicians. In the 

 present paper Poineare gives a new method 

 cl great universality for proving the so- 

 called Dirichlet principle. It depends 

 upon the fact that the boundary problem 

 can be exactly solved for the sphere and 

 also upon the theorem discovered by Green 

 that a potential function due to attracting 

 masses lying within a closed surface may 

 be exactly imitated by placing the masses 

 in a surface distribution on the surface of 

 the sphere. This Poineare calls the 6a- 



layage of the sphere, the masses being 

 swept out of the interior and deposited on 

 the surface. For any surface to be treated 

 the space within is filled up by an infinite 

 number of spheres such that any point 

 within the given surface lies in at least one 

 of them, and these spheres are swept in a 

 certain order so that the process is a con- 

 vergent one. The principle of Dirichlet is 

 thus established, but a practical method of 

 finding the solution is not given. The 

 other equation considered in this paper is 

 Fourier's equation for the conduction of 

 heat, 



- = «-A^- 



In this case the boundary condition is not 



as simple as in Dirichlet's problem, but we 

 have at the surface, 



are 



where h is called the emissivity of the body. 

 In this case it is demonstrated by the aid 

 of the calculus of variations that the prob- 

 lem is possible, the demonstration being 

 that of the existence of an infinite series of 

 functions Tin satisfying the conditions that 

 on the surface of the body 



and in its interior 



AZ7„ + fc,jU- = 0, 

 where the numbers k^, h^-'-kn are positive 

 constants such that 



\ < fc < fcs • ■ • . 



Phj^sically these functions have the prop- 

 erty that if the temperature of the body 

 at a given instant is distributed according 

 to any one of them then this distribution 

 will remain unchanged during all subse- 

 quent time, merely dying away at an ex- 

 ponential rate. It is interesting to notice 

 that in the last part of this paper Poineare 

 compares his process to that used by 



