420 PROF. A. C. DIXON ON STURM LIOUVILLK HARMONIC EXPANSIONS. 

 Thus the error in X 1 ' J <f> (b, a) is at most // 



(l+ M ){r|x| 1/a +|X I - 12 }exp (>-), 

 and the same may be proved for the errors in 



+ (?>,), *(&,a), X 1 *(/>, a); 

 also we may put Xi, XD for b, a. Hence, at most 



and or may lie as small as we please. If we make ra/f\~ l we have 



M//M-", 



so that in each of the four cases the error is of this order relatively to the true value. 



12. Since u, i\ U, V do not depend on a-, the error produced by neglecting or 

 altering a- is also of the order of X" 1 ' 3 in comparison with the true value. 



From (5) by putting p = //, a- = tr', and making X' approach X, we can deduce such 

 results as 



Jx~ * (b ' a} = J! + (x> a) * (x > b} dx > 1 



thus proving that <f>, i//-, $, ^, have everywhere finite differential coefficients with 

 respect to the complex variable X. They are therefore holomorphic functions of X all 

 over the plane. 



13. It is sometimes useful to know that <I> and i/r cannot tend to destroy each 

 other in such an expression as 3> + k\{s, where k is positive. 



To see this, begin at the other end of the interval (a, b). The argument of 8, 9 

 proves that the real part of \/ X?' (, b) Dv (x,b) is i | X | Q t sinh 2a (b x) where Q, 

 is positive and limited both ways. In this put x = a, and subtract from a multiple 

 of the result of putting b for x in (7). 



Since v (a, b) = U (b, a) and V (a, b) = V (/>, a) the real part ot 



x/^X {U (b, a) + kv (b, a)} V (6, a) 

 is thus found to be 



+ sinh 2a (*,-), 



where r , 1\ are the values of r at a, b respectively. 



Since k, Q, Q,, r , n are positive, and Q, Q u r , t\ are limited in both directions, the 

 ratio of U (b,a) + kv(b,a) to expa(6 a) is also limited in both directions. Combining 

 this with the results of 11 we have the following theorem: The values of 



\<t>(b,a), \{s(b,a), <t>(b,ci), ^(b,fi)/\/\ and aho &(b,a)+k\(r(b',a) irli<>r<- I: /* <t 



