. A. G. DIXON ON STURM -UOUVILLK HAI.'Mi >MC F.XP \NSIONS. 



427 



(o, I). Here the term "local average" means a function r constant in each sub- 

 iritrrval, and such that 



fl rl 



\p-r\dx and 

 . ji J ii 



1 __1 



P r 



dx 



tend to zero when the greatest of the sub-intervals does so. If - is limited, r may 



be the actual average of p in cadi sub interval. 



Tl xpansions discussed by LIOUVII.LK and STI I;M are those for which in the 



present notation 



I. = 0, GH = EK. 

 Thus 



K(t,x)~ 

 and since 



= when iJ = 0, 

 the typical term may be written as a multiple of 







K^ {.r, 0) + Jfy (j; 0) or K^ (.r, 1 ) - G0 (.'", 1 ) 

 iiiditU-iriitly, that is, it satisfies the equations (:{) and is such that 



when .r = 0, KI> = H^>, 

 when .r = 1 , K<I> = (!</,. 



and 



20. It may also be proved that if f(;r) is continuous and is equal to the sum ot the 

 former series ( 18), the new expansion holds also. For if Vi(t,x, R) is the special 

 function ~F(t,r, R) in which H, G, E, L are zero, that is, the function denoted by 

 F (/, ir, R) in 15, we have, after some reduction, 



^ 



*, ..... (16) 



which, even in the unfavourable case when K = 0, is finite when R -><, unless x, t are 

 each equal to one of the limiting values 0, 1. 



Now the difference of the two expansions for f(x) is 



that is. 



Lim - 

 H-> 



Liin ~- \f(t) { F (t, .*-, R)-F, (t, x, R)) <lt, 



f(x)\ {F (t, x, R)-F, (/, .r, U)} dt, 



for when the function to be expanded is constant, each of the expansions holds. 



3 I 2 



