FUNCTIONS IN THE THEORY OF IMKCKAL EQUATIONS. 193 



Tims we see that 



* (s} ~ L fr /l ( "-' K/ ' ( ' s+ ' )+/1 <-*-'>+/. <-' +e )] 8i 



converges uniformly to zero in (0, TT). 

 It may be shown in a similar way that 



sin ye 



has the same property, the function h l (s) being defined for all values of* by the rules 



/i, () = w ()/() (Osssw), 



= (-7T<*<0), 



It follows at once that, in order to establish the result stated, it will be sufficient 

 to show that, as n increases indefinitely, each of the four integrals 



. .- ..... (42) 



converges uniformly to zero in (0, TT). 



23. The function w(n) is defined in the interval (0, TT) only, and, by the hypothesis 

 of III., 1, has a continuous differential coefficient in this interval. Let us define it 

 for values of s in ( TT, 0) in any manner consistent with the conditions (1) that in 

 (IT, IT) it shall be always positive, (2) that it shall have a continuous differential 

 coefficient in ( TT, TT), and (3) that iv ( TT) = w (TT), /( TT) = w (TT) ;* then let us 

 define it for values of s outside ( TT, TT) by the rule 



The function w (s) defined in this way assumes only positive values, and has a 

 differential coefficient which is everywhere continuous. Also, on referring to the 

 definitions of the functions j\ (), /*, (,s), it will be seen that 



*,(#) = w (a) /,(), 

 for ah 1 values of s. 



* OHO method of doing this is as follows : Let "\ (.<) be the (possibly, a) rational integral cubic function 

 of .< whose coefficients are such that w\ (0) = w (0), Y (0) = w (0), Ci ( - ir) = to (w), w\ (-*) = w' (r). 

 'I ( - T) and Wi (0) are lx>th positive, we can clearly choose a number C great enough to ensure that 



-,() = ;, 



is positive in the whole of ( - -, 0). The conditions (1), (2) and (3) will all lie satisfied if we define u> (0) 

 to be equal to w 2 (s) in ( - w, 0). 



VOL. CCXI. A. 2 C 



