426 PROF. A. C. DIXON OX STI'UM l.K >r\'IU,K 1 1. \ UNION 1C KXI'AXSIOXS. 



',x, R) and group it with <lt in the integrations, that is, if \\c think of 



'' 



as the variable of integration in such expressions as 



\f(t)F(t,.rJi)dt and | F(f,.r, R)<//. 

 1 !>. The same method can lie applied when F (/, x, R) has the more general value 



where 



E, G, H, K, L are real constants, with the one proviso that when K is 0, GH is 

 positive or zero, so that the terms in Q involving G, H cannot tend to destroy each 

 other ( 13). Q cannot vanish except for real values of X if GH EK L 2 is zero or 

 positive,* a condition which includes the proviso made. 

 Thus it still follows that when < x < 1 



f(x) = Lim . 



J V ' H^x2rJo 



. 



(14) 



the summation referring to all the values of X for ichich 



S2 = 0, . . ......... (15) 



these values being taken in ascending order of magnitude. 



Sufficient conditions for the validity of this expansion are those already stated, 

 namely, that f(x) should be continuous at x and of limited total fluctuation ivithin 

 some neighbourhood containing x as an internal point. 



The restrictions placed on the functions a-, p in the fundamental equations (3) are 

 that 



(i) p shall be positive ; 



(ii) The integrals of \ a- \ , p, shall exist ; 



P 



(iii) A local average of p shall have a logarithm of uniformly limited total 

 fluctuation for some method of division into sub-intervals of the fundamental domain 



* For this and other results see 'Proc. L.M.S.,' ser. 2, vol. 3, pp. 86-90, and vol. o, p. 420; the former 

 passage contains a discussion of the case of equal roots. 



