and Electromagnetic Theory. 343 



i.e. 6'=— 2rA 2 /A,, whatever the function is, because the 

 apparent case of exception ^p s f(p)^~P i f\P) =0 is one which 

 makes the transformation nugatory. In a similar way if 



i*=p{l + e"), we get e'^-rBg/Bj. 



Again, if this function is used the estimate of the rate of 

 conversion to mechanical work at the surface, instead of 

 depending on rn in the factor 1 — rn, must depend on the 

 first-order term mf{p(l — rn)\. The main term in pressure 

 would then be proportional to 



pY(p)fr(l+A*)dn; 



or, if we were dealing with p 2 f(p)dp instead of p 3 f(p), it would 

 be proportional to 



ff{p)dp$n 2 {l + A?)dn. 



Thus on the geometrical side we should have to deal with 

 the same fraction J + A 2 or § — B 2 ; but this would be a 

 fraction of p z f{p)dp, the energy varying as p 2 f(p)dp; or the 

 whole ratio of pressure to the energy-content of the incident 

 wave would be 



Pf(p) ( 2 B) 

 f(p) ( * B2) ' 



When the fourth power is used 



f(p) =p, or pf(p) :f(p) = 1 ; 



but in any other case the energy converted would be a different 

 fraction of the energy-content in different parts of the periodic 

 scale. Thus, so far as first-order work is concerned, when 

 ic/V is small, the choice of a different function does not alter 

 the collective relations; but it does alter the way in which 

 pressure is related to energy-content, i e. it implies a difference 

 of efficiency as regards the production of mechanical work 

 by pressure, depending on the period of vibration. 



§16. There are some quantities continuous at a surface of 

 separation which have not received attention. 



(i.) KZ and M7 are continuous; i. e., for reflexion and 

 refraction Z 1 -fZ 2 =KZ 3 . The continuity follows from the 

 differential equations ; but in view of any doubt possibly 



arising in connexion with the forms -7- and yy, an independent 



proof for the reflexion of a wave is given. It may be remarked 



