BOUNDARY PROBLEMS AND DEVELOPMENTS. 



127 



may be deduced, whereupon it follows that 



T l f / N ^ 



= n7T~-J vPv T when x =F «, & 4= o, 



S ( m 4) Cr) 



2iri 



'ia> 



, —J {- (S^fl " JF 6 $f) F(6-0) -+MJ^ 



C 9 



J7Tt C 



M< 



when x = a. 



Z7TI c A 



when x = b. 

 Defining the matrices ^4 and Ki by the equations 



(119) 



-1 



Ka= L ) I3f (aj*»>) q^> H' 6 $f >) 



( UP 41 — 1 



'.IT 



it follows, therefore, that 







(5[-f) ^ ir 6 (5i-f ) , 



when x =(= o, .r =£ & 



(120) lim S l £(x) • j = X 4 F(6 - 0) • when 2 = a 



= K* F(b - 0) • when x = b. 



L 



A summary of the various results as contained in formulas (110), 

 (113), (118) and (120) is seen, in virtue of (105) and (111), to yield 



lim S m (x) ■ = \ Fix - 0) • + \ F{x + 0) • when x^a,x^b, 



TO— OO 



lim SM-= ± F(a + 0)- +K s F(a + 0)- + Ki F(b-0)-, 



TO" OO 



lim S m (b)-= ±Fib-0)- + K- i Fia+0)-+K i F(b-0)-, 



TO=0O 



In consequence we have the following 



Theorem : Given that L is any vector differential system of the 

 type 



