DR. J. MERCEK: STURM-LIOUVILLE SERIES OF NORMAL 



1 _' 



\\ll.MV 



When p increases indefinitely ft clearly converges to ; and therefore the right-hand 

 member of (24) converges to zero. It follows that 



r"/(-0 dt iEE? p \ e- ft f(s-t] dt. 



Substituting t = x, (w) in the right-hand member, and then replacing w by t, we see 

 that this may be written 



JO 



^ P 



-xi (01 x'l (0 dt - 



Taking this in conjunction with the result previously obtained, we see that (22) will 

 IK? established when it has been proved that 



_ 

 pf 



Jo 



12. We have 



-Xi (01 x'. (0 *- 



'i (0 



- ( 2G ) 



Now, by a known theorem of the differential calculus, 



where, as above, , is a point of the interval (0, t). 

 From this we see that 



lxi(0-l=sK (o 



and hence that 



Thus the right-hand meml^er of (26) is 



when; 



^ 



a f ' fe !/[- W] !'() *. 



