PROF. A. C. DIXON ON STURM-LIOUVILLE HARMONIC EXPANSIONS. 413 



These integrals are indefinite, each carrying with it an additive constant tc \- 

 detexmioed by trial of some particular \alueof.r. The fonnnla (5) includes a great 

 variety of particular results of which many arc well known in the theory of the 

 equation (l). 



Km- instance, take <f>, />' to It- $(x,a), </> (.',/') so that p = p', X = X', a- = <r ; by 

 putting .f = a, h in turn we have 



tf> (/>,) = -tf> (",/>), 

 and similarly 



,/, (/,, ) = * (a, b), * (/,, ) ^ -* (,r, />), 

 and 



(r, ) <I> (.1-, tt) ^ (a-, a) * (., a) = 1 . 



Again, taking 0, ^' to Ije (.r, n), ^'(a:, />), we have 



, a) <l>'(.r, h) + (<r'-,r + \/p-\'/p') 0(.r, ) ,/,'(.'-, 6)} cfa (6) 



which is the formula to be used in the approximation. The right side of (<j) is. the 

 error committed when $'(!>, a) is taken as equal to ^ (/*,); in it the term depending 

 on <r'<r turns out to be unimportant, while when X' = X the rest is numerically less 

 than the square root of 



f' (,,-,/)' ^ 



. (i 



f* 



It will be our object to choose // so that (pp')*ds is small, while at the same 



. n 



time 0' can be expressed in terms of known functions, and, in fact, the interval (a, b) 

 will be divided into small sub-intervals in each of which // will be constant. Thus we 

 need the following lemma: 



If f(x) is a function limited and sumnutble in (n,f>) (hi* interval can be so divided 

 info a finite number of sub-intervals, and a function <j>(x) constant in each sub- 



('' 

 iufcrrnf can be no chosen that (f'' fa') 3 <!<' is arbitrarily mnnf/. 



Jn 



5. To prove the lemma, let U, L be the boundaries of /(.?); take n 1 ai'ithmetic 

 means a t , a y , ..., _! between them and let o = L, a m = U. Let the values in (, b) 

 fur which a r -i <.f(.c) Sa r form the set l r (r =0, 1,2, ..., n). Enclose l r in a set of 

 intervals A r , not overlapping, and C'(/ r ) in a set P r , not overlapping, so that A r , \\ 

 have a common part < e. Let the intervals of A r in descending order of length be 

 ^i, <Va> , and take an integer p such that 



Z ,V> A r - t (r=0, l,a, .... n). 



