BOUNDARY PROBLEMS AND DEVELOPMENTS. 



55 



whence, substituting from the system 



y-j = 2 a ik y k j, i, j, = 1, 2,...w 

 k=i 



(which is equivalent to equation (1)) we obtain 



d\ Y 

 dx 



2a u - y k \. . .Sflu-?/fc, 



k k 



2/21 



■l/2n 



l/nl 



■Vnn 



+ 



2/n 2/m 



2 a2kVki- ■ .2 aohVkn 



k k 



llnl 



2/nn 



+ ...+ 



2/11 

 2/21 



■2/ln 

 •2/2n 



2 a nk yki- ■ -2 a nk y kn 

 k k 



d 



i.e. - | y I = a n | F | + a 22 | Y | + . . .+a nn | F |. 



Let us suppose, now (i), that the elements of A(x), Y(x), and Y'(x) 

 are each continuous in an interval a ^ x ^ 6, and (n), that at some 

 point a: of this interval | F | =£ 0- Then throughout a neighborhood 

 of the point in question 



d\Y\ 



— — — = 2 a^fc da-, 



\ I \ fc=l 



or, integrating, 



| F | = ce 



/ 2 a a dx 



Inasmuch as the right-hand side of this equation cannot vanish if 

 c d£ 0, it is seen that the hypothesis that | F [ =£ for some x leads to 

 the conclusion that [ F | differs from zero for all x. Thus we infer the 



Theorem: If F(.r) is a matrix of functions which satisfies equation 

 (1), while | F(.i-o) [ = 0, a ^ x ^ b, then | F(.r) \ = 0, a ^ x ^ b. 



A matrix of functions Y(x) of this type which satisfies equation (1) 



