430 PROF. A. C. DIXON ON STURM LIOUVILLK HARMONIC FA'PANSIONS. 







with these values of a n , b n , since it may be verified that when A = A,, 



(.r) (t) { Ky, ( 1 , 0) + H0 ( 1 , 0) \ -.= ( I ) n ( " ) ('' (>, ') + ( L - &) '/ (.'', } , 

 and theivfoiv. by addition. < (t,x) is the same as 



but for a constant factor, since 61 + 62 = 2L. (Compare ' Proc. L.M.S.,' ser. 2, vol. 5, 

 p. 473.) 



24. The integral (20) may now be written 



= 1 .Op 



a form which shows that it decreases continually as m increases, and therefore tends 

 to a definite limit when m increases without limit. 



Also 



fri ^ 



<lj>* \\ - n ('i') 'In (') 



( P 



i -j ri ^ 



= - /'(') f (.'< ; ) dx x -/(*) ^,, (x) dx 



.0/0 Jon 



Jo Jo pj.pt 

 which shows that the integral (20) is equal to 



I ' I (fx y dx+ J_ I 1 f Lf( x )f(t) F (t, x, II) dx dt 



JO p Z(7T Jfl Jit p z 



if Jl is so chosen that the path (II) encloses \\, \ 2 , ..., X,,. 

 Since then 



Lim | F(t,x,li)dt = - 



o 



the limit to which (20) tends when R is indefinitely increased is the limit of 



-^-\ l \ l -(Jx-ftYV(t,x,}i)dxdt ..... v - (21) 



4(7T JO JO p x 



25. From this form it is possible to prove that the limit is zero. 



First, let the domain (0, l) be divided into intervals in each of which /(.<) is 

 constant. Then when x, t are in the same interval the contribution to the double 

 integral (21) is zero. When x, t are in different intervals (ic > ^i), (fn, t\),fx ft has a 



