92 Lord Rayleigh on the 



and J x is defined by 



z / z 2 2 4 z 6 \ 



J ^ = 2V 1 ~274 + 2T4^6"*2.4 2 .6 2 .8" f,, • > )• ^ 47) 



If cos-J%=0 (i. e. in the direction of original propagation), 

 (46) becomes ira 2 , every element of the area acting alike. 

 This is the maximum value. When % is such that 



2&acosi%=7TX 1*2197, 



the secondary light vanishes, at a greater angle revives, then 

 vanishes again, and so on, the angles being of course func- 

 tions of the wave-length. If we conceive the cylinder to 

 increase in size gradually from zero, the scattered light 

 vanishes first in the backward direction % = 0, in which direc- 

 tion evidently the greatest differences of phase occur. Every 

 thing is determined by the course of the function J x ; and (46) 

 within the limits of its application embodies the theory of 

 Young's eriometer. 



We will now consider the case of an obstacle in the form of 

 a sphere. If z be a coordinate measured perpendicularly to 

 the plane containing the primary and secondary rays, formula 

 (46), multiplied by dz, will represent the effect of a slice o 

 the sphere, whose radius is a and thickness dz, and what 

 remains to be effected is merely the integration with respect 

 to z. For this purpose we write z=c sin <£, a = c cos <f>, where 

 c is the radius of the sphere. The integral then takes the form 



kc0S L y \ Ji(^ccosix cos ^) cos2 ^#' • (48) 



or, if we expand J x by (47), and integrate according to a 

 known formula, 



27tc 3 f 9 m? 

 ~3~r 5" + 7T 



5.4 9.7.5.4.6 

 m 8 



+ 



11.9.7.5.4.6.8 



}, m* 



in which m is written for 2kc cos J %. It will be understood 

 that (49), after multiplication by e int AK~\ gives merely the 

 value of P in (36), and that to find the complete expression 

 for the secondary light in any direction other factors must be 

 introduced in accordance with (35), (37), (38). The angle ^, 



* July 15. — I tind for the first root of (49), ?»=4 - 50, giving as the 

 smallest obliquity (jr— \) at which the secondary light vanishes, 



7r-x = 2sin- 1 (4'50/2*c). 



