of Reflexion and Refraction. 293 



9. We shall now investigate the laws of reflexion and re- 

 fraction at the surface (supposed to be plane) of the transpa- 

 rent substance to which the aljove equations apply, taking for 

 granted the results arrived at in the memoir (Cambridge 

 Transactions, vol. viii. p. 7) above-quoted, and assuming the 

 notation there employed (observing that n=-k v=k''i/). 



10. Referring to section viii. (p. 24), we find that the equa- 

 tions of motion and co7inection^ are satisfied by the following 

 imaginary vakies of V, V^ and V (the vibrations being parallel 

 to the plane of incidence), viz. 



^'-^5 + 5' + >3 2^^ ^ 



H* l]a.s + s'-f»J J 2 

 where jw, = -y = — (since n — kv =■ k' t/), 



and p ss [jt,p'f or k' p' = kp ; 



V is the velocity of propagation of transversal waves in va- 

 cuum, and if is given by equation (5.) of the present paper; 

 p' and q' are tiierefore imaginary. 

 Let us assume 



A;'s' = /:((r — x\/^), 

 then, since kp = k' p' and^^ + s'^ = Ij we find 

 k^p^ + F(<r .^\/^^)^ = /c'2 



w 



-C'n\/-1 

 ~ B' 



and ,,p^ + (<, _ ;^ ^31)2 = I (l - ^ ^^)' 



by (5.) 



observing that 



F = ^ = 5-. 



,2 «i2 



7^* w^ 



11. If the vibrations be perpendicular to the plane of inci- 

 dence, the following are the corresponding values of V, Y , and 

 V, viz. 



* The equations of connection obtained in section ii. are evidently the 

 same, whether the particles of matter exercise forces of resistance or not ; 

 for the forces of resistance are of the same order of magnitude as the forces 

 which appear in the equations of motion, and therefore they do not appear 

 in the equations of connection. 



