100 On the Electromagnetic Theory of Light. 



ary disturbance approximately vanishes in the direction of the 

 primary electrical displacements, agreeably with what has 

 been proved before. It should be stated here that (67) is not 

 complete to the order &V in the term containing cos 0. The 

 calculation of the part omitted is somewhat tedious in general ; 

 but if we introduce the supposition that the difference between 

 k' 2 and k 2 is small, its effect is to bring in the factor (1 — ^Pc 2 ). 

 Extracting the factor (k /2 —k 2 ), we may conveniently write 



16 



,2 



- rns 



8 



c cos 2fl], . (QS) 



in which 



n rC C ~v K C rZ C" ^ n 



COS V ^ 5- COS 20 



lb o 



= cos 0- V 2 c 2 -k 2 c 2 _ B<? cog2 ^ t > (69) 

 16 4 



In the directions cos = 0, the secondary light is thus not 

 only of high order in kc, but is also of the second order in 

 (k'—k). For the direction in which the secondary light 

 vanishes to the next approximation, we have 



^-e=^(i^-hV)=^^^. . . (70) 



This corresponds to (61) for the sphere; and is true if kc, k'c 

 be small enough, whatever may be the relation of k! and k. 

 For the cylinder, as for the sphere, the direction is such that 

 the primary light would be bent through an angle greater than 

 a right angle. 



If we neglect the square of (& /2 — & 2 ), the complete expression 

 corresponding to (69) is 



cos 0(l-ip c 2 )-i k 2 c 2 cos 2 = cos 6[l-ik 2 c 2 -ik 2 c 2 cos 0]. 



This may be compared with the value obtained by the former 



method, viz. cos J 1 {2kc cos \ 0)-i-kc cos \0, and will be found 



to agree with it as far as the square of kc. 



If we suppose the cylinder to be extremely small, we may 



confine ourselves to the leading terms in (65) and (67). Let 



us compare the intensities of the secondary lights emitted in 



the two cases along 0=0, i. e. directly backwards. From (65> 



k !2 c 2 -k 2 c 2 

 ^cx 2 > 



while from (67) , 7 „ „ k' 2 — k 2 



