418 PROF. A. ('. IMXON ON STURM -I.IorVll.I.K HAUMOXIC KXI'AXSloXS. 



product of all the factors > I and II a that of all the factors < I among tin- 

 quantities T*. 



If log r is of uniformly limited total fluctuation then HIT" and II : ,r"' arc limited, :ind 

 so, therefore, are P, Q. 



y. Now 



= PJX I cosh 2oc (-), 



and heu.ce |Xr 2 | and [Di'l* are severally less than this. 

 Also, the real part of y--\vDv is 



oa 

 = iQ|x|sinh2a(a,--) ....... (7) 



Hence \v\ and jDr -r v\\ are both 



<{Pcosh2a(a a)} 1 '' 2 , 

 but 



sinh 2a (x-a) {P cosh 2a (./-)}""' 2 - 



Now P, Q cannot increase or decrease indefinitely, and therefore, if we do not 

 allow a to tend to zero, we have that \v\ and |Dr -f- \/\ \ bear to expa(.r a) ratios 

 which are limited in both directions ; and this is true, both when the sub-intervals are 

 increased in number and also when | X | is increased indefinitely. 



In the same way it may be seen that the ratios of |\/X and jDu| to exp (.' ft) 

 are limited in both directions. 



When a = the argument shows that j\/X|, Du\, jt'j, and Dr -r- v/XJ are 

 limited above, but not below, and, in fact, we know that each of the four is capable 

 of vanishing. 



10. The condition that log r should be of uniformly limited total fluctuation is 

 necessary, for if it is not fulfilled P may be indefinitely great or small, and it is 

 conceivable that u\/\ and D?<, for instance, shoiild become very small together, in 

 the same way that a pendulum would be practically stopped if its velocity wen- 

 suddenly reduced in a constant ratio at every passage through the lowest position 

 and increased in the same ratio at every time of reaching one of the extreme 

 positions, when, of course, the increase would be of no effect. 



11. Hence if in (5) we take the limits to be a, b and put />' = r, a local average 

 of p, o-' = 0, X' = X, <l> = (./, a), <{>' = u (.r, b), we have 



= <f>(b,a)u(b,a) 



(.,-, , ( ) ,/(.,-, !,)-<,$ (.<, a) (.,-, b) 



