256 



This represents the deflexion y, (a.); it is seen to be the same as 

 the quantity y, of I, 5. In the same way we continue and so we 

 may prove that (9) agrees term by term with the series of paper I. 



7. In case the expansion do not converge, it may happen that 

 the method of graphical integration, communicated in paper I, 

 remains still valid (vid I, 7); this depends on the approximate 

 proportionality of the functions y„i^) for large values of ?j. We 

 shall prove this now ; more exactly : we shall prove 



km = — n 



11-^00 .Vri (•») 



where fi is independent from x. 



Now K„ (.<", s) is represented by the absolutely and uniformly 

 convergent series 



i^ / t\ ^ "'"' (*) ""'" (^) 

 '»=! A" 



where the quantities A,„ represent the numbers i^i'iliY and (^?i//)* in 

 the order of their magnitude and the functions iüm{^) are the 

 corresponding normal orlliogonal functions. Putting 



nun(%)q(%)d^+ ':E Q\w,„{èi) = P„ (10) 



ƒ 







we get the absolutely and uniformly convergent series 



°° Pm V>m («) 

 y„{x)=(-ky-y 2 ' {n=\,2, . . .) 



m = 1 ) 



''m 



Supposing h to be the smallest value of m for which P,„ ^ 0, we 

 can write 



The series in the right hand member of this equation has an 

 absolute value which is less than the sum of the series 



■^ Pm tOm (x) I , 



1)1 ^ 1 *A+m 



a quantity which is independent from n. From this and from 



Urn (—]=0 



we get 



