in the Undulatory Theory. 167 



It is important to bear in mind that this solution has been 

 obtained solely from the conditions of elhptic polarization, the 

 original equation being in the forms (10.) (11.) retaining all 

 its terms. 



(6.) If instead of the formula (9.) we had taken the ex- 

 pressions for the component displacements rj f , as not in- 

 volving such a relation, but simply as in the original expres- 

 sions (1.), or supposing b = m the forms (6.), we might 

 still pursue steps analogous to those above exhibited, though 

 with different values. To trace these results we have only to 

 alter these formulas agreeably to the new conditions. Thus, 

 for unpolarized light, on making sin 6 =0 cos b = 1, we 

 have 



[ f n^ S « 1 



from (21.) =^ \-m S [«p 2 sin» ff]J (28.) 



L -m S [/3 J 2 sin2 6'] 



from(22.) 0={ I l^tl^ (^^^ 



lVom(2.) 0={ I V^tl^ (SO.) 



from (23.) = ^ \-mX [/3 p' 2 sin" ^] J (31.) 



1^ —m 2 [a g 2 sin^ 6] 



Now in all these equations it is evident that since « and /3 

 are by the original condition nsohoUy arbitrary and independent 

 both of each other and of the other quantities, these equations 

 can only hold good for all values whatever, oi en. and /3, if each 

 of the terms involving respectively a. and /3 are separately 

 = 0, that is, we must have 



from (29) 1^ = ^ t«^ '^" ^ ?^ ^^^-^ 



rrom ^IJ.) ^^ = S [^ ? sin 2 6*] (33.) 



from rsO) /0 = S [/3p'sin2^] (34.) 



trom (30.; ^q = S [« ? sin 2 6*] (35.) 



from (28.) = S [/3 y 2 sin^ ^] (36.) 



from (31.) = 2 [« <? 2 sin^ 6*] (37.) 



from (28.) = n^ S u-m 2 [ap 2 sin^ ^] (38.) 



from (31.) = w-2/3-m2;[(3 2j'2sin'-^]. [2,^.) 



Hence the formula for unpolarized light is only a solution, 

 provided those conditions are fulfilled in the original equa- 

 tions. 



