MR P. ALEXANDER ON THE EXPANSION OF FUNCTIONS. 319 



But 



A = 



ra 



j <p.G n .x x dx 



/V 



(23) 



00 ctoc 



Hence from (21) and (22) this becomes 



_ ft*[G„ . cr(8+g'»)- 1 0] a , =<i 



d 



v. dg n ) x=a 



(24) 



which verifies (14) for this case. 



If X = and p = (Fourier's Heat, ch. vii. and viii. ; Eayleigh's Sound, 

 § 135), then (23) or (24) will give an expansion of (f>(x) in linear functions 

 (trigonometric), 



$(x)=A .^—cos(x l jg )+A 1 ^/—cos(xJff 1 )+&c. 



where g Q , g v &c, are the roots of 



a Jg . tan (a Jg) = ah . 



If X = l (Fourier's Heat, ch. vi.; Eayleigh's Sound, § 201), 



4>{x) = A . J p (x ,Jg Q ) + An . J p (x JgJ + &c, 



where g , g v &c, are the roots of 



JgJ' p (a Jg)+hJ p (a Jg) = , 

 which gives an expansion of <f>(x) in cylindric or Bessel's functions. 



* Since writing this I have proved that if 8 = X 2 (-j-j + X X — + X 



A, --* 



f^dx 

 " e 2 

 4>.G n - — v dx 



A, 



Gn- — v dx 



but I find that Sturm and Liouville have anticipated me (Liouville's Journal de Mathematiques, vol.'., 

 1836). 



