100 BIRKHOFF AND LANGER. 



It can, moreover, be easily shown (see page 107) that when the 

 system (46) is simply compatible at every characteristic value, this is 

 not true for k= I, i.e. 



J -Z {l \x) R{x) Y {l) {x)-dx + 0, for any /. 



a 



Let us suppose now that an arbitrarily chosen vector F(x) • can be 

 developed into a series of the form 



(77) F(x)- = £ c k Y™(x)-. 



fc-i 



Multiplying both sides of this equation on the left by the vector 

 •Z {l \x) R(x) and integrating term by term we have formally 



b b^ 



J-Z {l \x) R(x) F{x)-dx = £c,J -Z {l \x) R(x) Y {k \x)-dx, 



which in view of relation (76) reduces to 

 b b 



(78) 



f-Z (1) (x) R(x) F(x)-dx = cj -Z {l \x) R{x) Y {l \x)-dx. 



Inasmuch as the matrix on each side of this equation is one all of whose 

 elements are identical the equation may equally well be written 



b 



I 2 z { ?(x)y h (x)f h (x)dx [ (1) 



6 



c t < I £z%>(x)y h (x)ynx)dxya), 



a 



whence 







ci = — b 



f 'izV\x)y h (x)yi l) (x)dx 



a 



