I I NOTIONS IN THE THEORY OF IXTHJKAL EQUATIONS. 14J 



For, with the hypothesis stated, there exists a finite number v such that 



. . . , ... . (38) 



for all values of n* The ratio of the numerical values of the corresponding terms of 

 (36) and (37) thus tends to unity as n is increased indefinitely. It follows that (37) 

 is absolutely convergent. 

 Again 



For negative values of X the numerical values of the terms of the series on the right 

 are less than the corresponding terms of 



= i X M X 

 Since we have 



it is thus clear, that, as X tends to oo, the left-hand member of (39) will converge 

 uniformly to zero, if 



= i X B X 



has this property. But 



S D^iff = K, (s, ,), 



II = 1 A B \ 



and therefore steadily diminishes to zero as X tends to oo.f It follows from DIM'S 

 theorem that K x (s, s) converges uniformly to zero, for values of * in (0, IT). Hence 

 the left-hand member of (39) converges uniformly to zero in the square s s :s IT, 

 ^ t s IT, and the lemma is established. 



18. From this it appears that the difference between (36) and (37) satisfies the 

 requirements of the theorem of I., 3. Hence the difference between 



< 40 > 



converges to zero, as X tends to oo, uniformly for values of s in (0, IT). It follows 

 that the results obtained in 13, 15, 16, remain true when 



is replaced by (40). We have thus established the theorem : 



* PirfIV.,8. 



t "Functions of Positive and Negative Type and their Connection with the Theory of Integral 

 Equations," 'Phil. Trans. Roy. Soc.,' A, vol. 209, pp. 443, 444. 



