BOUNDARY PROBLEMS AND DEVELOPMENTS. 



101 



Consequently we have 



00 



b 



/n 

 2 tf>(. 

 h-l 



x)yh(x)fh(x) dx 



(79) F{x)-= I { 



f S «!*>(*) 7*(a) vPto dx 



-(*) 



F w (a;)' 



If, therefore, F(.t) • may be developed into a series of form (74) which 

 converges in such a manner as to legitimatize the processes above, 

 then (79) is a necessary form for the series in question. 



Thus far it has been stipulated only that Y (x)- and -Z \x) be 

 respectively solutions of system (46) and its adjoint for X = X*,. But 

 each of these systems is homogeneous, and if Y (x) • and -Z (x) are 

 any particular solutions then c Y (x)- and c-Z (x) are also solutions. 

 Having chosen a definite pair Y (x) • and -Z ( (x) we have then 



F(x) 



00 



z 



b 



2 CZh 

 h=l 



(x)y h (x)f h (x) dx 



/n 

 h=i 



-(k) 



cz^(x) 7h (x)y^(x)dx 



Hk) 



Y^(x) 



If in particular c is chosen so that 

 b 



cf^-zi k \x)y h (x)yi k \x)dx= 1, 



h=\ 



and the vector c-Z , for this value of c is associated with Y • so that 

 the choice of one implies the choice of the other, we have, on dropping 

 the bars over the letters, 



1 



Y {k \x)-. 



(80) F(x)- = L 



fc-i 



f2zi k) (x)y h (x)f h (x)dx 



In using this formula it must be remembered that -Z (x) is deter- 

 mined as soon as the particular 1 (x)- is chosen. 



