318 

 The 



Mr. R. Hargreaves on Radiation 



integrated relation is 



rr' + 3 arc sin / 



a 



=p /3 [V (1 + r 2 ) + ^37r — arc sin /— rr'X], 



where r*= \ 7 1— r 2 . In the limit when r' is small or w 

 nearly = V, p /3 /p z varies as r' h or (1 — r 2 ) 5 / 2 . The approxi- 

 mate form analogous to (7) is 



and the exact estimate o£ pressure is 



^ (arc sin /+ r/) + — (V -- arc sin r' + rr'). 



7T 7T v ' 



In one dimension we have the single relation 



p(l-w/V)=p'{l + u>/Y), 



which must be duplicated in the form 



jp«(V-«7)(l-tp/V)=|/ a (V '+w)(l+tc/V), 



to show the change of period and velocity of transit. No 

 integration occurs, % and %' are written for p 2 and £>' 2 , and 

 the pressure is % + %'. 



The special features in the comparison of the cases for 

 different dimensions are : — for radiation formulae 



Jx 



2 ' 



2V X 



IT 



and V% ; 



for the limits when w = Y or r=l, p' varies as (1— ?') 3/4 , 

 {l-rfl\ 1-r; and tf as (1-r) 3 , (1-r) 5 / 2 , (1-r) 2 . 



§ 4. When refraction accompanies reflexion there are two 

 channels into which energy is directed, and the two trans- 

 formations concerned are only part of the problem : the 

 partition of energy requires distinctive optical theory. The 

 special features affecting the transformation for refraction 

 are (1) a modified velocity of propagation in the dielectric, 

 and (2) the necessity for interpretation with reference to the 

 moving standpoint of the dielectric. 



At first we use an argument referred to the same co- 

 ordinates as the original wave, take p" for period number, 

 and suppose H the velocity of propagation to depend on 



