with a concentric core of a different material 405 



5. In § 2 it has been assumed that the equation (9) has no 

 imaginary roots or repeated roots, and that it has an infinite 

 number of real roots, symmetrically placed with regard to the 

 origin. 



There is no difficulty at all in proving that the equation has no 

 pure imaginary roots, and that its real roots are not repeated; 

 and, since F {a) is an odd function of a, to every positive root a, 

 there corresponds a negative root — a. 



It is harder to prove that there is no root of the form ^ ± cq, 

 where ^ and t] do not vanish. To prove this, we need to show that 

 the equation 



a cot aa + cot aa ih — a) ^ — - = 



fxaa 



has no root of this form, when /a, a, a and h are real and positive 

 and h > a. 



Let ?7i = sin «r, < r < a "l 



, jj sin fia{b~r) . ,> (17). 



and (J 2= ~ ,1 X sm aa, a < r < o\ 



sm /xa {o — a) ) 



Then ^' + a^U^ = and '^^ + fi^a^U^ = (18). 



Also 

 Ui = 0, when r = 0: U^^ JJ^, when r = a: U.^ -^ 0, when r = b 



(19). 



Further, if a is a root of our equation (14), 



dU.^ J J ( dUx 



dr 



— U2= fJ'f^ [a —r^— UA, whenr =^ a ...(20). 



Now let a, ^ be two roots of equation (14). 

 Also let Ui, U2 be as above, and let F^, V^ be the corresponding 

 expressions, when ^ is substituted for a. 

 Then 



(«2 - ^2) / [" u^v, dr+^r U,V,dr) 



= Wu,V:'-V,U:') dr + - f (C/^Fa"- F2C/2") dr, by (18) 



.'0 fJ-f^ J a 



U,V,'-V,U,' 



0^ /XCT 



U,V,'-V,U,' 



= (C/iF/- FiC/i') - -^ {UxV,'-V,U,'), when r = a, by (19) 



= 0, by (20). 



But if a and ^ are conjugate imaginaries ^ ± 117, Ui and Fj 

 are conjugate imaginaries; also U2 and Fg are conjugate. 



