202 DOUBLE KEFBACTION AND CIRCULAR POLARIZATION 

 If we set A 2-n-A 7 



~2<7T p Z ' 



and 



the equation reduces to 



fri'f* 



~27T 2 ' ^ 



(14) 



(15) 



1 + e &yi+/yii+#] 



aa 



+ ey 2 2 



X (16) 



p- 



where a, b, c, e, /, g are constant, and a quadratic function of 

 L, M, N, for a given medium and light of a given period. 



13. Now this equation, which expresses a relation between the 

 constants of the equations of wave-motion (1), will apply, with those 

 equations, not only to such vibrations as actually take place, but also 

 to such as we may imagine to take place under the influence of 

 constraints determining the type of vibration. The free or uncon- 

 strained vibrations, with which alone we are concerned, are charac- 

 terized by this, that infinitesimal variations (by constraint) of the 

 type of vibration, that is, of the ratios of the quantities a 1} (5 lt y x , 

 a 2 , /3 2 , y 2 , will not affect the period by any quantity of the same 

 order of magnitude.* These variations must however be consistent 

 with equations (4), which require that 



L da^ + M d^ + N dy l = 0, L da 2 + M d/3 2 + N dy z = 0. (17) 

 Hence, to obtain the conditions which characterize free vibration, 

 we may differentiate equation (16) with respect to a 1? /3 lt y 1? a 2 , /3 2 , y 2 , 

 regarding all other letters as constant, and give to da l} d^ 1} dy v 

 da 2 , c/3 2 , dy 2 , such values as are consistent with equations (17). 

 Now da l} d/3-L, dy l} are independent of da 2 , d/3 2 , dy z , and for either 

 three variations, values proportional either to a lt /B^ y x , or to a 2 , /3 2 , y 2 , 

 are possible. If, then, we differentiate equation (16) with respect to 

 a i> & Vi an d substitute first a lt /3 l} y x , and then a 2 , /3 2 , y 2 , for 

 da l} d/3 l} dy l} and also differentiate with respect to a 2 , /3 2 , y 2 , with 

 similar substitutions, we shall obtain all the independent equations 

 which this principle will yield. 



If we differentiate with respect to a lt /3 lt y 1} and write a lt /3 15 y 



for da 



dy l , we obtain 



l} l , 

 * 



aa 



[L(/3 iy2 



) + M ( 7l 2 - a iyt 



(18) 



Compare 11 of the former paper, page 189 of this volume. 



