904 



SCIENCE 



[N. S. Vol. XXXVIII. No. 991 



Fourier in deducing his equation; namely, 

 by supposing the body to be composed of 

 a large number of small bodies each radia- 

 ting heat to all the others according to the 

 law that the amount of heat radiated in a 

 given time is propoi'tional to the difference 

 of temperature of the two bodies. Thus a 

 system of ordinary differential equations 

 is arrived at, 

 dVi 



at 4=1 



Vk)+CiVi = Q, t = l, 2---71, 



which is readily solved by putting 



Vi = Uie->^', 

 in which ease the differential equations be- 

 come algebraic linear equations for the 

 quantities TJi, 



\Ui = :S,Ca{Ui~ Uk) +CiUi. 



k 



In order to solve them it is necessary that 

 the determinant 



Ci — X, — Ci2, — Cn, 



Cl2) Cs — X, — C23) 



A = 



should vanish. But we get the same equa- 

 tions if we consider the quadratic form 



* = 2CaiVi - VkY + ^CiVi\ 

 which being equated to a constant repre- 

 sents an ellipsoid in n-dimensional space. 

 The equations for the axes are our linear 

 equations. The axes of this ellipsoid being 

 all real, all the roots of the determinant A 

 are real. The form may then be decom- 

 posed into a sum of squares. 



^ =^ \(j>± -\- Xii05° • • • , "where 



Upon the properties of this quadratic 

 form depends the whole theory. When 

 the number of particles becomes infinite 

 the system of ordinary differential equa- 

 tions leads in the limit to Fourier's partial 

 differential equation, and the theorems 

 which will arise if the passage to the limit 

 is justified lead to Poincare's deduction. 

 It is to be noticed that this principle had 

 been used before by Lord Rayleigh in con- 



nection with the theory of vibrations and 

 the possibility of passing to the limit postu- 

 lated by him is now known as Rayleigh 's 

 principle, ilore interesting still is the fact 

 that to-day this process used by Rayleigh 

 and Poincare has become in the hands of 

 Fredholm and Hilbert a rigorous method, 

 that of integral equations, which is at pres- 

 ent occupying a large part of the atten- 

 tion of the mathematical world. 



In his second paper on the same subject 

 published in 1894 in the Eendiconti del 

 Circolo Mathematico di Palermo, Poincare 

 passes to the consideration of the more gen- 

 eral equation 



AM-t-f«+/ = 0, 



where f is a constant and / a given space- 

 function. This equation includes not only 

 Fourier's equation but the equation of 

 waves 



if we put 



■ e'"'!*, { = 



Regions in which / is not zero are called 

 sources of heat or sound. Poincare pro- 

 ceeds in this equation to develop u accord- 

 ing to powers of ^, as had been done by 

 Schwarz, thus obtaining a solution by suc- 

 cessive approximations which he proves to 

 be convergent. He also proves the funda- 

 mental property that a solution of the 

 equation is a meromorphic function of the 

 parameter i having an infinite number of 

 simple poles, that is to say, 

 iUi 



V = -Z, 



■ki 



where AUt + hUi = 0. 



This theorem is also fundamental in the 

 theory of integral equations. To speak in 

 the language of sound Poincare demon- 

 strates the existence of an infinite number 

 of natural vibrations for the air in a cavity 

 surrounded by the given surface, the char- 

 acteristic numbers ki or values of the poles 



