Ufidulatori/ Theory of Light — Absorption. 183 



Put ae^^ zs a, 7it + kx + h — (/i, then >] = a sin w, ^ = /3 « 

 sin [cti 4- y) = /3 a (cos y sin co + sin y cos m) ; which last ex- 

 pression for ^ gives 



(^ — j8 « cos y sin w)^ == (/3 « sin y cos co)^ 

 = (/3 a sin yY (1 — sin^ co). 

 But, since a sin «; = >), this equation gives 



(?— /3 cos y, >])- = (/3 a sin yY — (/3 sin y. ij)^: 

 hence we find 



^"" I ^' 2cosy.v]^ _ 



an equation to an ellipse of which y^ and ^ are the coordi- 

 nates. 



Consequently, when the system is in the state of motion 

 expressed by the equations (43.) every molecule describes an 

 ellipse round its place of rest ; and the equations (35.) show 

 that the general motion of the system is equivalent to a num- 

 ber of coexisting motions of the same kind. 



The period of the revolutions of the molecules, in the 



2 TT 



movement represented by (43.), is equal to — • ; where 2 tt is 



the circumference of a circle whose radius is unity. And 

 this movement is transmitted through the medium in a series 

 of continuous waves ; the length, or rather thickness, of each 



2 TT . 



wave being — . The direction in which the waves travel 



depends on the sign of k, supposing that of n to continue 

 the same. But, by the equations (22.), it appears that -the 

 sign of 11 is arbitrary : therefore n as well as the arbitrary 

 quantities a and b, may be written either positively or nega- 

 tively. Now if we change the signs of «, «, b, in (43.), it is 

 virtually the same thing as changing the signs of k and y, 

 while those of a, 7«, b, remain the same. Consequently, when 

 we take for the positive direction of x, that in which the 

 waves travel, Ave may write the equations (43.) thus: 



>j = a e^ ^ sm. {nt — kx + b)^ 



t, = ^ a e^"" s\n{nt - k X + b - y) , ^^^"^ 



and suppose w and k to be positive. 



The intensity of the light is considered to be measured by 

 the vis viva of the molecules, which, when other things are 

 equal, is proportional to the square of the amplitude of vibra- 

 tion. Thus, when the movement is represented by (45.), the 



