70 BIRKHOFF AND LANGER. 



i) The elements of G(x, t) are continuous in x except for x = t. 

 Along this line (x = t) there is a jump of unit magnitude in the 

 elements of the principal diagonal, i.e. 



G(x, x - 0) - G(x, x + 0) = I. 



ii) For any given t, G{x, t) satisfies equation (6) in x, except along 

 the line x = t, i.e. 



dx 



iii) For any given t, G(x, t) satisfies the boundary conditions of 

 system (6) in x, i.e. 



W a G(a, t) + W h G(b, t) = 0. 



Conversely we have the 



Theorem: The dependence of G(x, t) upon the variables x, t is com- 

 pletely determined by the characteristics (i) to (iii) above. 



Proof: Suppose G (x, t) is a matrix possessing the characteristics (i), 

 (ii), and (iii). By (i), (ii), J(.r, t) defined by J(x, t) =G (x, t) - G(x, t) 

 is continuous for all x, a ^ x ^ b. Moreover, J(x, t) satisfies (iii) and 

 is therefore a solution of system (6) in x. But system (6) is incompati- 

 ble by hypothesis. Hence J(x, t) = 



and G(x, = G(x, t). Q. E. D. 



It is readily verified that a further set of three characteristics which 

 completely determines the dependence G(x, t) upon the variables may 

 be obtained by interchanging x and / and replacing system (6) by 

 system (7) in the discussion above. 



The fact established by this theorem should be carefully noted. 

 While the choice of the pair of associated solutions F/, and Z/, on the 

 right-hand side of equation (12) is not unique, yet the entire function 

 G(x, t) is independent of that choice. 



