PROF. A. C. DIXON OX STUBM-LIOUYILLE HARMONIC EXPANSIONS. 



415 



i, a a , ... be the points of a subdivision AI, according to A, for which the value of the 

 integral is < e. 



Take a subdivision B,, according to B, in which the sum of the intervals containing 

 ai, a 2 , ..., a n is < e : then in tin- other intervals of B, the value of <p.r is constant 

 whether in AI or BI and the integral can therefore IM> made < * , while for the 



intervals containing a, a, the value of the integral is not more than (U L)'*. 



Hence for the subdivision B! the integral can IKS made lews than 



1 1 lilt is, arbitrarily small. 



Hence the subdivisions of (a, 6) may IK- taken all equal, or according to any other 

 method, provided that the greatest of them tends to zero. 



Moreover, the square of 



\'fx-tf>x\dx 



.'a 



is less than 



f* 

 (I i a) (Jxdacf doc, 



.'a 



and, therefore, 



f'' 

 \f.c-<f>x\dx 



Ja 



is also made arbitrarily small. 

 If L is positive, then 



i: 



_L .J_ 

 ./> <!> 



'(.' 



and is also arbitrarily small. We may, in fact, say that 



is arbitrarily small. 



This is the property that will l>e immediately useful. It may l>e extended to an 

 unlimited function fx when 



r* 



f f 



I \f.i-\dx and 



.* a .' f 



dx 



exist. For take a limited function f t x which is equal to f.r when N i (f-<'Y 1/N 

 N being a certain positive number, and is 1 for other values of x. N may be so 

 chosen that 



f* 



l/ r ~ f \x\dx 



J a 



and 



t 1 



d.r lx)th 



Then the function tj>x, constant in each of a system of sub-intervals, can l>e so 

 determined that 



