400 REPORTS ON THE STATE OP SCIENCE. i 



subject to the condition 



\u>{x)]Hx=l 



a 



is r- where X x is the smallest value of X for which the homogeneous 

 integral equation 



b 



$(or) - \L(x,t)<l>(t)dt = 



a 



can be satisfied. The other roots A 2 , A 3 are obtained by finding 



the maximum value of the integral when a number of additional 

 conditions of the type 



U(t)t(t)dt 



a 



are introduced, f r (t) being solutions of the homogeneous equation for 

 tbe roots A,. 



Hilbert assumed in his investigation that k(x,t) was a definite func- 

 tion, so that all the quantities A,, are positive. This restriction has 

 been removed by the author, 1 who has found limits within which the 

 double integral must lie ; the method depends upon a use of the energy 

 function. Another proof of Hilbert's theorem is given by Holmgren. 2 



There is an analogous theorem due to Dirichlet for differential 

 equations, and this has been used by Mason 3 to establish the following 

 general theorem : — 



There exists an infinite series of normal parameter values A„ and 

 corresponding solutious (normal functions) y n of the differential 

 equation 



y" + \Ay = (1) 



and the boundary conditions 



y(a) = 0, y{b) = . . . . (2) 



If A changes sign in (a,b) the values A„ include an infinite series of 

 positive terms \ 1 <A 2 <X 3 < . . . ., increasing without limit, and an 

 infinite series of negative terms 



A_ 1 >A_„>A_ 3 > .... 

 decreasing without limit. The function y n satisfies the conditions 



b h 



(" ky^lx = ± 1 f ky.ydx = 



"[»- = ±1,±2, . . . ±(«-l)] 



(in which the upper or lower signs are to be taken according as n is 

 positive or negative), and gives to the integral 



1 Camhr. Phil. Trans., 1008, vol. xxi. 2 Comptes rend its, 190.". 



3 Avier. Trans., 190G, vol. vii , p. 337. 



