FUNCTIONS IX Till: THEORY OF 1NTKCKAI. Kol .\TK>NX 133 







3. In what follows we shall consider in detail onlv tin- case when the pair of 

 boundary conditions for (a, h) is B. A slight modification of the method developed 

 below is necessary to obtain a formal proof when the pair of conditions is a B, *B, 

 or *B, but the nature of this modification is so obvious that we shall content 

 ourselves with a statement of the corresponding results. 



After what has been said in the previous paragraph, it is clear that the problem of 

 di ti nnining the solutions of (l) which satisfy B is equivalent to that of determining 

 the solutions of (2) which, for suitable values of p., satisfy B'. 



It may happen that certain of these values of p. are not all positive. If so, we can 

 choose u numl)er K which is less than the least of them. The equation 



is then such that the values of fi for which there exist solutions satisfying B' are all 

 positive, and clearly the aggregate of these solutions is identical with the aggregate 

 of the solutions of (2) which satisfy B'. It follows that, without loss of generality, 

 we may suppose the values of /u, for which there exist solutions of (2) satisfying B' to 

 be all positive. 



It has been shown by KNESER that the GREEN'S function* of 



- ........... <"> 



for the pair of boundary conditions B' is 



ic (M) =*(*)*() (*<) 

 = *()*() (*). 



where 0(*)t satisfies (3) and the Ixnmdary condition 



$-/ t '=0 at x = 0, 

 as 



</ (.<)t satisfies (3) and the boundary condition 



^ + H' = at- * = IT, 

 as 



and the two functions are chosen in such a way that 



* For the theory of GREEN'S functions, see HILBERT, 'Gott. Nachr.' (1904), pp. 214-234, and KNESER, 

 Math. Ann.,' vol. 63, pp. 482-486. 



t From a theorem on linear differential equations it is known that, as q u 



exist and have continuous second derivations in the interval (0, T) (ride PICARL, -Traite" d'Analyse,' 

 tome III. (1896), p. 92). 



