Magnetic and Electric Energies as a Pressure 665 



the pressure is still given by the value of L, as other terms 

 on the right hand vanish. A component of tangential stress 

 is nowX z -w(Yc~Zb)IV or -Z(X-w&/V)-c(d + wY/V), 

 which vanishes. 



To deal with the separate waves write X x + X2 for X, and 

 take (6) in conjunction with 



Pih=P^2^ Pi(^ T — wn \) — Pz(y + W7hi)i 

 jt? 1 m 1 =jt? 2 ?w 2 , pi(Vn 1 —w)=p s (Vn2 + w), ... (9) 



which bring the arguments p^li.v + m 1 y + n 1 z—Yt) and 

 pi{h< v + m 2y — n i z ~^ r t) into agreement for z — wt = constant. 

 Using (6) with multipliers p 2 l 2 , P2 m ^ we £ e ^ Zi = Z 2 ; then, 

 quoting (5), 



EiW-fii 1 ) = dHZ/^' + Z,' - E 2 (l-» 2 2 ), 

 and E 2 = E lj? V/^i 2 (10) 



connects the energies of incident and reflected waves. With 

 multipliers X! Y l5 (6) yields 



SX 1 X 2 =-E 1 p 2 / i ; 1 + 2Y 2 Z 1 7(V 2 -^). . . (11) 



Then L = 2«i« 2 — SXjX 2 



= (J\h+ /»im 2 — ?2]»2 — ljSX^g- (/ 2 X!+ WljYx 

 — » 2 Z 1 )(/ 1 X 2 + ??i 1 Y 2 + 7?,Z 2 ) 



= {-E 1 p2/>i + 2VZ 1 2 /(V 2 - ? ^)}{-2(V» 1 - ? ,) 2 



x i ) i/y ) 2(^ 2 -^ 2 )} + ) i ?? i+^2^2) 2 Zi7/h^2. 



Writing y>i»i+/>2 ? *2 = V(/ ? i ~ p*)\ w from (9), the Z, section 

 vanishes through 7> 2 72 — w 2 ) = />i(V 2 4- id 1 — 2Ywn 1 ), and 

 finally 



L = 2E 1 (V,* 1 -^) 2 /(V 2 -z<; 2 ). . . . (12) 



If the calculation of pressure is made through the Z' z 

 formula (modified), the demonstration of the vanishing 

 of product terms is the troublesome part. It follows from 



X 2 -wb 2 /Y + X^l-vmJV) + wl 1 Z 1 jY = 0, ) 



a 2 \ ivYz/Y + Y^n.-iclY) -m{L Y = 0, J 



and associated formulae. The parts of the pressure are given 

 by square terms, viz , 'E 1 iT 1 {n 1 — wjY) and E 2 n 2 (n, 2 + ivjY), 

 with ratio pii^ : p 2 n 2 . 



Phil Mag. Ser. 6. Vol. 39. No. 234. June 1920. 2 X 



