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From tliese equations A, B, C and D are easily solved; putting 

 A, {q) = cosh ^ (jl sin \ qI -{- sink i qI cos ^ qI, 

 A, (q) = cosh i qI sin ^ qI — sinh ^ qI cos i qI, 

 we get 



— 4 ^' A; (q) i A cosh Q {x—h I) + C cos q (.i- ^ I) \ = 



= {eosh \ qI cos ^ qI -\- sinh ^ qI sin ^ qI) cosh q {.v — i /) cosh q (g — ^ I) 



-|- cosh Q (x — ^ I) cos Q (§ — ^ I) -\- cos Q {x — | I) cosh (§ — ^ I) 



-{- (cosh 4 qI cos i ol — SMi/i 4 p/ «M ^ oZ) co« y (.r — ^ Z) cos q {§ — ^ Z), 



— 4 q' A, (p) I 5 smA Q [x—h [) + DsinQ{x-^l)\ = 



=i (cosh ^ qI cos 4 qI — sinh ^ qI sin A qI) sinh q (x — ^ *'"^ ^l5 ~ è 



+ sin/t p (.« — k Ï) sin Q {^ — ^ i) + sin q {x — ^ I) sinh q (§ — ^ Z) 



-)- (co«A \ qI cos ^ qI -\- sinh ^ qI sin ^ (jl) sin Q {x — k I) sin p (i — ^ I). 



We now have calculated the function K{a;,^,).); it appears to be 

 a function with the denominator 4^' A, (p) A, [q). Tiie values of X 

 equating to zero this denominator are the characteristic numbers of 

 the problem; as K{x,§,k) is symmetrical with respect to x and § 

 that numbers will be all real. From this it follows that the cor- 

 responding values of q have an argument that is a multiple of 

 ^ Jt ; it is easily proved to be an even multiple so that the values 

 of p wil be real or purely imaginary and the corresponding values 

 of A negative or zero. For that purpose we first write 1 — co.s/iqIcosqI 

 for 2 A, (p) A, (p) and then substitute in it p/:^«-j-t/i; equating 

 the real part to zero we get 



cosh a cosh ^ cos a cos ji -j- sinh u sinh (i sin a sin c? = 1, 



which is not satisfied by |J = =t « ^ 0, for substituting ^ ^ ± a in 

 it we get sinh' a = sin'' a, wliich is impossible for u^O. Therefore 

 the values of p are real or purely imaginary and the characteristic 

 numbers are negative, except one which is zero. 



If p be a root of A, (p) ^ 0, also <p will be a root (and con- 

 sequently — p and — jp); the same is true with respect to the 

 roots of A,(p)=0. We now call the positive roots of the equation 



tghp = — tg p, 



in the order of their magnitude /»,,/), and the positive roots 



of the equation 



tffh p = tgp, 



ordered in the same way q^,q„... Then the characteristic numbers 

 will be 



