BOUNDARY PROBLEMS AND DEVELOPMENTS. 103 



But since -Z (t) and Y (x) • are vectors 



Y {k \x)--Z {k \t) = ( S *{»<*) tf\t)\ = n (y\ k) (x) zf(t)). 

 Hence we have from (81) 



(83) gfeO-E^M ••*"«>. 



Having obtained this formal development of the Green's function 

 it is possible to state the 



Theorem: If the development (S3) converges to G(x, t) in such a 

 manner that a uniformly convergent series is obtained by multiplying 

 (83) on the right by any matrix of continuous functions and integrating 

 term by term, then any vector F(x) • the elements of which are con- 

 tinuous and have continuous first derivatives, and which satisfies the 

 boundary conditions 



W a F(a)- +W b F(b)- = 



is represented by a convergent development of the type (80). 

 Proof: The relation 



F'(x)- = B(x)F(x)- +C(x)- 



defines the vector C(x) ■ , and inasmuch as F(x) • is then a solution of 

 the non-homogeneous system 



Y\x)- =B(x) Y(x)- +<?(*)• 

 W a Y(a)- +W b Y{b)- = 0, 



it is given by the formula 



b 



F(x) ■ = J G(x, t) C(t) ■ dt (see page 20). 



a 



Substituting for G(x, t) its series development we have 



6 



„, v f? Y^(x)--Z w (t)C(t) 

 F(x) = J L "T- ~ dt, 



fe-i nX 



