308 Mr. R. B. Sangster on some Consequences of 



and let AB represent the section of a plane wave-front 

 incident on Fl, the section through the plane interface 

 bounding the second medium, in the figure supposed the 

 denser. The edge B is just touching the interface. Through 



Fig. 1. 



A draw AC perpendicular to AB meeting FI in C. Let 

 fi = AC, then with centre B and radius BD = 1, describe the 

 arc HG. Through C draw CD touching the arc HG at D, 

 and join BD. In the progress of the wave AB into the 

 second medium, AC is the path of the extremity A, and the 

 disturbance due to the extremity B reaches D in the time A 

 reaches C ; therefore, DC is a momentary position of the 

 wave-front in the second medium, while 



BD : AC : : 1 : /* (1) 



Also, the areas ABC and BCD are generated in equal times, 

 and we can assume unit measure for that dimension of these 

 areas which is perpendicular to the paper, hence 



area area 



S : S' : : ABC : BCD 



: :BA'AC :BD'DC. ... (2) 



But, the angle ABC = z= tan _1 yu, (by hypothesis). Hence, 

 tan ABC = /t/l = AC/AB ; and BD = 1 when AC = ^ (1) ; 

 therefore, AB = BD. But AB and BD are sides of right- 

 angled triangles standing on the same base BC, hence area 

 ABC = area BCD, and S = S'. 



We have now to find expressions for sin i and tan i when 

 ^S = S', where % may be > or < 1 but is always positive, 

 and it may be stated here that this application and interpre- 

 tation of x i s adhered to throughout the paper. 



