ON THE THEORY OP INTEGRAL EQUATIONS. 



401 



w-fa)'* 



its least possible value consistent with these conditions and the equations 

 (2). This minimum value of J is ± A„. 



More general theorems have been deduced for the differential equation 

 and more general boundary conditions. 



These problems arise naturally in the Calculus of Variations in the 

 following way l : — 



Consider the problem of determining a curve y = y(x) joining two 

 given points for which the integral 



= \F{y',y,x)dx 



has a minimum value. 



Let y = y(x) be the equation of the required curve and 



y = y + eM 



a neighbouring curve, where u is supposed to satisfy the following 

 conditions : 



(1) In the interval a<x±b, %i{x) possesses a continuous first 

 derivative. 



(2) u vanishes for x — a and for x = b. 

 We then have 



a 



+ |V 2 Fn+>'«F l2 + ^F 22 ) + . 



= 3 + Ju(F 2 -^)dx 



a 



.]dx 



+ 



+ 



where F,, F^ F,,, . . denote the partial derivatives of ~F(y',y,x) with 

 regard to ?/ y. 



Now since y makes J a minimum 



F,-i(F,) = 



and 



u'°-F u - m 2 iM^ - F 22 ) is positive. 



Hilbert now considers the problem of determining the function u(x) 

 so that the quantity 



b 



fi = [ C«' 2 F U - u* ( f lJ 12 - F 22 )~| dx 



a 



1 Hilbeit, Lectures on the Calculus of Variations, Gottingen, 1904-05. 



