318 Mr. R. Hargreaves on 



be taken as zero to form the system written in (3 a). The 

 form of (3 a) and the connexion with the divergence equa- 



tion-^ h ^ — =0 for Z = 0, show that <r is a stream 



3* By . . 3^ 



function in two dimensions. The condition <^- on the 



oy 



barrier gives X = as well as Z = 0, i e., tangential electrical 

 action vanishes. For sound waves where yjr is a potential, 

 this is the condition of no velocity at right angles to the 

 barrier. 



Also the use of ^jr = cos Jc(Yt + x sin a + ?/cos a) in (3a) 

 for the incident wave, makes 



c=— &sin&( ), X = — k cos « sin k ( ), Y==/csin asin&( ). 



The electric vector is therefore perpendicular to the axis o£ 

 z and of amount 



Y sin a — X cos a = £sin k(Yt-\-x sin a + ?/cos a). 



A second group has 



Here the condition — -*- =0 gives a zero value for tangential 



electrical action on the barrier, while for sound it gives zero 

 pressure. The electric vector in the incident wave is 

 parallel to the axis of z. 



To deal with an arbitrary plane of polarization, we 

 may write ^ — cos 7 cos k(\ 7 t + # sin a. + y cos a) in (3 a), 

 sin y cos & ( ) in (36), and add the values of XYZ derived 

 from the two solutions. For the incident wave then 



X= — k cos a cos 7 sin k(Yt + x sin « +y cos a), 



Y = k sin a cos 7 sin k( ), Z = — k sin 7 sin k( ). 



§ 2. Where the incident wave is 



yjr — cos k (Yt -f Ix + my + nz), 



there cannot be complete independence of z as in the preceding 

 work. It is a matter of intuition to perceive that the role of z 

 is limited to its phase-effect. The mathematical expression 



of this limitation is that ^- = ^ ^- in the equations (1) 



