428 PROF. A. C. DIXON ON STURM LIOUVILI.K HARMONIC KXI'ANSIOXS. 



In this expression the parts depending on F, FI are separately negligible by 



HOUSON'H theorem, except for values of t between x + n, and for such values 



/(') f(*) is s>!ill and F FI is finite. The whole is therefore arhitrarily small and, 



in fact, zero; thus if the one expansion is valid for f(x), so is the other,* provided 



that f(x} is continuous at .r. 



21. Again, the error produced in one of the functions ,/,. \>,, <|>, ^ by neglecting or 

 altering a is relatively of the order of X" 12 , and thus it follows that if F, F, are 

 functions of the present type ( 19) the same except for a change in or then 

 Y(t,x, R) FI (t, u", R) is limited. Similarly, then, the expansibility of f(x) is not 

 affected by a change in the value of o- in the fundamental integral equations (3) so 



long as ||o-|(x exists and f(x) is continuous at x. 



22. In order that the more general expansion may hold at the limits 0, 1, some 

 further conditions are necessary. It does not seem worth while to discuss these in 

 detail, but they are satisfied 



(1) When /(I) =/(0) = ; and 



(2) When /(l) =/(o) and G = H == L = 1, K = E = 0. this being the case of 

 a periodic function expanded in a series of periodic terms, since 



(17) 



is the condition that functions <j>, < may satisfy the equations (3) and have the 

 same values at both ends of the interval (0, l). 



On account of the periodicity there is no occasion to distinguish between the two 

 end-points or between these and the other points of the domain. 



23. It is known that the Fourier constants , b n of a function f(x) are such as to 

 give the least possible value to 



{f(x} cos nx 2,b n sin nx}" dx 



J * 



and there is a similar theorem for the present expansions. 



The condition that there may be functions (x), (x) satisfying (3) and fulfilling 

 the boundary conditions 



. - (18) 

 is 



'(1,0), 



h This result and that of 21 were first given for the usual LKM vi.i.i.i. scries \>\- .1. MK.KCKR (sec Roy. 

 Soc. Proc.,' A, vol. 84, pp. 573-5, and 'Phil. Trans.,' A, vol. U11, p. 147). 



