178 



Assume 



Dr. J. Hopkinson on the Effect of 



(p =c x sin\#,~l 

 yjr = c 2 sin \x> J 



(5) 



where c x , c 2J and X may be imaginary, but <£ and t|t are real : this 

 form is indicated as suitable, because f must change sign with a?. 

 We obtain 



whence 

 and 



where tan = km, 



a q (c { — c i km)X 2 =rn 9 c v "1 

 6 



cos o 



^: 



The most general real expression for <£ is then 

 • ■A 1+ B,v^T siB ^ 1 + v - Ttan |^. 



or, as it may be written, 



$ = A 1 sin^a? 



e /xtan 2 ' _|_ e -fttan 2 



Lx 1 



Similarly 



4- Bj cos/^ . 



T|r = A 2 sin /x# 



4- B 2 cos /X^7 



2 





6 6 

 utan-. x — utan-.# 

 6 2 — £ 2 





2 





e | l tan-.*. + e - M tan i 



a? 



2 





e Mtan-..r _ e ~^tan 2 



a? 



The constants will be connected by the relations 



fA 1 + By-l= A a V31_B 2 , 



l Al -By^l = -A 2 V"=l_B 2 ; 

 that is, 



A,= -B 2 and B, = A 2 . . . 



(6) 

 (7) 



(8) 



(9) 



