306 Lord Rayleigh on the Propagation of 



introduce the simplifications arising out of the largeness of 

 the arguments in Q and Q 2 . 



If z be a complex quantity of the form % + ir), we have in 

 general 



cos z = cos | cosh 7) -~i sin £ sinh 77, . . . (30) 



. sin2f + i sinh2?7 , 1X 



tan^= -— f-— (31) 



coszf + cosn 2t] 



Thus, when rj is large, 



d log cos z 



: = —tan z — —1; 



clz 



so that when y=yi, since h is a pure imaginary, 



The introduction into (26) of these values and those of X! 

 and A, 2 from (29), gives 



d log Q , _ y'/fi 

 dy a 2 



wdiere 



y= N / At / + (a/5-6/a) N /j/; .... (33) 



or, if :=y 1 \^{m 2 -\ 1 '), 



ztanz = y'hhjja 2 (34) 



This is the equation by which z, and thence m 2 , is to be 

 determined. 



In the case corresponding to that treated by Kirchhoff 

 y v is not so large but that the right-hand member of (34) is a 

 small quantity. The solution of (34) is then 



« 2 = 7'%iA' 2 ; ..... (35) 

 whence 



m z = Xl + ^=-Jl+~^\. . . . (36) 

 a?y x a 2 \ y 1 sjh) 



We now write h = ni, so that the frequency is nftir. . Thus 



•/*= >/»») . (l + i) (37) 



and 



m= ±(m' + im"), (38) 



where by (36) 



(rf'"-^. »"=-H~^/- • • (39) 



Vl.zayi a s /z.zay 1 v ' 



