THEORY OF THE DISPERSION OF LIGHT. 261 



The examination of the cases thus arising, is closely connected with the question 

 of the evanescence of the terms before alluded to (15.). These relations therefore 

 we shall now proceed to trace. But it will be desirable first to point out more clearly 

 the nature and meaning of those conditions. 



(13.) It is evident from the value of r (8.) that it does not change its sign along 

 with a change of sign in A a?, A y, or A 2 ; the same is true therefore of <p (r) and 

 %// (r). It is also evident that ^y^, A %^, do not change the signs. But the product 

 A 3/ A z, will change signs as each value oi I^y corresponds to a negative and a posi- 

 tive value of A %. 



Hence the terms 



-^ (r) sin A; A JT 

 %//(r) Ai/^sin /r A J7 

 "4/ (r) A z^ sink Ax 



-^{r) Ay A z sink Ax 



'^{r)AyAzsm'^y—^) 

 will have values with opposite signs. But, 



^|/(r)Ay2sin2(-^) 



"4/ (r) A ^2 sin^ \—q-) 



will have all their values with the same sign. If then we consider the sums of a 

 number of such terms respectively, those of the first set may become = if the 

 positive and negative sums be equal. In those of the second set this cannot happen. 

 Hence resuming our abridged notation we may have either 



2 (;? sin 2 = 

 S ip' sin 2 ^) = 

 2 (q sin 2 0) = 



or 2 {p sin 2^) = s 

 . . (36.) 2(ysin2^) =^, j^. . . . (38.) 



2 (q sin 2 ^) = s^,. 

 2(^sin2^) = 0. . . . (37.) "2 (q sin^ 0) = s,„ (39.) 



but always 



2 {p sin2 0)=s! 

 2 {p' sin2 0) 



The conditions which give these values respectively, are dependent on the general 

 supposition relative to the arrangement of the molecules. If the distribution of the 

 molecules in space be uniform, the values (36.) (37.) obtain. If not uniform, the 





