368 REPORTS ON THE STATE OF SCIENCE. 



function G(x,£) which satisfies certain boundary conditions and whose 

 (n — 1)"' derivative experiences a sudden change in value of magnitude 

 — 1 at the point x = £. In the case of a differential equation of the 2nd 

 order, the Green's function can usually be expressed in the form 1 



where u(x) is a solution which satisfies the boundary condition at one 

 end of an interval {a, b), v(x) a solution which satisfies the boundary 

 condition at the other end of the interval. If the linear differential 

 equation is L(«) = 0, then a solution of 



L(u)+f(x)=0 



which satisfies the same boundary condition as G and has a continuous 

 second derivative is given by the formula 



b 



u{x) = jGfoQflfyK 



and conversely if u(x) is given, a solution of this integral equation of the 

 first kind is provided uniquely by the formula 



f{x) = - L(u). 



The solution of L(w) + \u = which satisfies the boundary condi- 

 tions also satisfies the integral equation 



f(x) = xJcKavQAW 



and this determines the possible values of A. 



If the differential equation is self adjoint, the Green's function is a 

 symmetric function of x and £, and it often happens in virtue of the 

 formula (2) that it is a definite function. It then follows from the theory 

 of integral equations that the values of X are all positive. 



The question of the existence and nature of the different values of 

 X for differential equations of various types has been discussed very 

 thoroughly by Mason who uses a method depending on the calculus of 

 variations. 2 



Extensive applications of Green's functions to problems of mathe- 

 matical physics have been made by Picard, 3 Hilbert, Mason, Plemelj, 

 Boggio, 4 Lauricella, 5 Hadamard, 6 and many other writers. The 

 literature on this subject is now very extensive, but a good idea of the 

 different problems may be derived from the articles by Bocher and 



1 See the papers by Hilbert and Kneser, Math. Ann., Bd. 63. Many examples of 

 Green's functions are given in these papers, and also in a work by A. Myller. Diss. 

 GUtingen, 1906. 



2 Trans. Amer. Math. Soc, 7, 1906, p. 337. 



* Liouville's Journal, ser. 4, 1890, p. 145, ser. 5, 1906, p. 129; Rend. Palermo, 

 1906 and 1909 ; Annales de VEcole Normale, 1908. 



* Rend. Lincei, 1907, p. 248. 



» Annali di Matematici, 1907, ser. 3, t. 14, p. 143 ; Rend. Lincei, 1908. 



* Memoiret de VInstitnt, 1909. 



