[ 411 ] 



XI On the .SVr/V.v O/STITRM and LIOUVILLK, as Derived from a Pair of Funda- 

 mental Integral Equations instead of a Differential Equation. 



/{// A. ('. UIXON, /"'./(. X. . f'rofessor of Mathematics, Queen's University, Belfast. 



Received April 6, Bead May 11, 1911. 



Introduction. 



THK series of LIOUVILLK and STURM are generally treated by means of approximate 

 solutions of the fundamental differential equation, these approximations being valid 

 when certain functions involved in the differential equation have differential co- 

 efficients. The object of the present paper is to relax this restriction, and for this 

 purpose integral equations are used in place of a differential equation, and an 

 approximation is investigated (4-ll) depending on a function which is constant 

 throughout each of a system of sub- intervals. 



In 15-18 the results are applied, by help of HOBSON'S general convergence 

 theorem, to that one of the Liouville series which is usually valid at the two ends of 

 the fundamental interval, and in 19-22 to the more general series discussed by me 

 in ' Proc. L.M.S.,' ser. 2, vol. 3, pp. 83-103. 



A theorem analogous to that of VALLEE-PoussiN on the series of squares of the 

 Fourier constants is then proved ( 23-25) by a method which I believe to be new. 



1. The differential equation of LIOUVILLE and STUKM is 



and is equivalent to the pair of integral equations 



ft) 



The values of U, V at the lower limit in these integrals are the two arbitrary 

 constants of the complete primitive. 



In the theory of the equation (l) the known functions- A', g and the unknown 

 function V are generally supposed to have differential coefficients ; no such assumption 

 will l)e made in the present treatment of the equations (2). All integrals will be 

 taken according to LKBESOUK. Also g, k are supposed positive and I real, and it is 

 assumed that the integrals of g, AT 1 , |/| exist. 



VOL. OCXI. A 481. 3 G 2 30.11.11 



