424 Sir William Thomson on the 



16. To verify that the sum of the activities (rates of doing 

 work per unit of time) of the four reflected and refracted 

 waves is equal to the activity of the incident wave, consider 

 pencils of them, all cutting the interface in a square with its 

 sides respectively perpendicular and parallel to the plane of 

 incidence. The activity of each of these pencils is equal to 

 twice the kinetic energy in a length of it equal to its wave- 

 length, divided by the common period ; or, which is the same, 

 twice its kinetic energy per unit volume, multiplied by its 

 sectional area, multiplied by its propagational velocity. Now 

 twice the kinetic energy per unit volume of a wave of either 

 species in an elastic solid is equal to the density of the solid, 

 multiplied by half the square of the maximum molar velocity; 

 and the sectional areas of our five pencils are respectively 

 cos i, cos/, cos {/, cos/,. Thus the activity of the incident 

 pencil, for example, is 



?.io, 4 (F//3) 2 cos7/3 (50); 



or, by (24), 



ico'^FH (51); 



and is similarly found for the others. Hence the activities of 

 the five pencils, each divided by ^eo 3 , are respectively 



£F 2 Z ... incident (distortional) (52); 



£Gr 2 1 . . . distortional reflected (53); 



J y l t ... distortional refracted (54); 



£C 2 \ . . . condensational-rarefactional reflected (55); 



£yCy 2 \y. . . condensational-rarefactional refracted (56) . 



17. The first of these must be equal to the sum of the other 

 four, and thus, subtracting from each side the second, we find, 

 as a form of the equation of energies, 



t*(F + G)(F-G)=#, + fC 2 \+?,C, 2 X, . . (57), 



which is verified by (47), (48), and (46). In verifying it we 

 find, from (46), 



? C*X + ?/ C,*X ( =^!!^g .... (58) 



which is an important expression for the sum of the energies 

 carried away per unit of time by the reflected and refracted 

 condensational-rarefactional waves. In using these results, 

 (52) . . . (58), it is convenient to remark that, by (24), (22), 

 (38), we have 



, co . co . . , co ( . /3y 2 . 2 \i 



£ cose; m —-o smv > h— -q\ 1 — "z>2 sm l ) - (59); 



