Manchester Memoirs, Vol. xlii. (1898), No. 3. 15 



The critical values of ka are determined by the 

 equation 



Q,oska-\ii\ka 2~* A~)sin/^a= +1 



This requires either that 



sin ^^ = o 

 or that 





(79). 

 (80), 



(81), 



or that 



cot^/&« = ^^^-^^) . . (82). 



The positive roots of (81) and (82) are determined 

 graphically by the intersection of the curves 



jj/=-tan^x, 2' = cotJ^ . . (83), 



with the hyperbola 



We have now total reflection throughout the range 

 extending from ka = o to /^^ = j3„ where (5o is the lowest 

 positive root of (81). For large values of ;ir the hyperbola 

 (84) approaches the asymptote j^ = J^iur, and the critical 

 values of ka approximate to those determined in § 4. 

 It will be noted that no special peculiarity attaches 



