Propagation of Waves in a Resisted Fluid, 331 



Multiplying (2.) by V — I and adding it to (1.), we find 

 !?!ff±|^ + ± (T+ N ^31) ^&H^- 



df^ ^ V ^ ' dt 



-^ ?^^ — 



(3.) 



10. To obtain a solution of this equation, we shall suppose 

 that the light is circularly polarized, i. e. that each slice has a 

 circular motion of translation parallel to the plane o£xy. In 

 this case N T and v (which in general are functions of/ and z) 

 will manifestly cease to vary with t; and their variation with 

 z also will be insensible compared with that of ^ and r|; for f 

 and *} go through all their values when z increases by a wave's 

 length, whereas the radius of the circle of vibration, and there- 

 fore N T and v, can suffer only an extremely small change for 

 so minute an increment of z, except the light be so rapidly 

 absorbed that the substance may be regarded as opake, and 

 not transparent, as we have supposed. Hence we may with- 

 out sensible error integrate the equation (3.) on the supposi- 

 tion that NT and v are independent of s as well as of/. 



Now, the vibrations being circular, the expressions for ^ 

 and ij must be of the form 



^=ucos(nt—kz), vi==usin{nt—kz), 

 where n and k are constants and u is independent of /. (We 

 suppose n and k here to be essentially positive, which amounts 

 to supposing that the wave is propagated in the positive direc- 

 tion along the axis of z.) 



Hence we have 



and if we substitute this in equation (3.), we find 



To solve this linear equation put, as usual, u=ae *^, and 

 we find 



-n^ + IL (T a/^~N) = A{h^ + 2kh\^^ — k^), 



