,,, DR. J- MERCER: OTUBM-LIOUVILLE SERIES OF NORMAL 



5. The singular values of *(., t) (III., 3) being X 1( X,, .... X., .... we have 



which may be written _^_ 



D (\)= n (i- 



where m has the signification of 2, and ft is any positive integer greater than 1-m, 

 which is such that (9) holds for == n. Again, the singular values of the GREENS 

 function of (10), for the boundary conditions (11), are the values of /. for which there 

 exist solutions of 



satisfying these boundary conditions ; they are therefore -*, I 2 -*, 2 2 -*, ..., n 2 -K, 

 Hence 



It follows that 



X 



D(X)_ .i _ p(x) 



A W" S /i- x 



.?,V (n-l) a 

 where 



n"KTn\/n 3 K 



P(x)=n ^.n(i-^g:^? r - - - (13) 



x .5\ W 2 -/c-X/\ T,, / 



Now from (8) we see that there exists a positive number v such that 



for all values of n sz. n. From this it is easily seen that 



/ n^if 



n i- n * 



and 



n 



n = n \ ' n 



T, 



are both convergent. Further, recalling that K is negative, we have 



-!. (14) 



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

 Equations," ' Phil. Trans. Royal Society,' Series A, vol. 209, p. 445. 



