64 Mr. Gwyther on a 



I thus get as the final form of displacement in the secondary 

 wave 



xyz . 27r 



4 = T-sin— (/'/ - /') 2X. 



The object of this paper is to study to what extent an 

 expression can be found to represent, on the basis of Huy- 

 ghens' principle, the action of a plane polarised ray of light. 

 From simple principles we may learn something of the way 

 in which the co-ordinates must enter such an expression so 

 that they may allow of resolution. 



Imagine a ray of plane polarised light to travel along 

 the axis of x and let its components be <3:cosa and a^m.a 

 along the axes of j/ and z, and let, if possible, the consequent 

 disturbance at a point x.y.z be represented by a dis- 

 turbance in the secondary wave, such that the part of it 

 arising from the component ^cosa along the axis of j/ may be 

 U = ^COSa(/i +/2J1/ +/3.S: ^-f^f- ■vfr.yz 4-/6^2 + &:c.) 

 V = acosa((])i + (p^y + ^gS + cp^y- + cp^yz + (p^z'^ + &c.) 

 W = ^cosa(-(//i + -ii^^y + ;//3S + -^^y"^ + -^^yz + \//6^^ + &c.) 

 when all the functions contain x and r only {y" + z^ being 

 always written r^ — x"^ whenever it appears) and the series 

 being continued in the same way. 



Hence the part which arises from <^sina along the axis 

 of z will be represented by 



U' = ^sina(/i +f,z -f^y ^W -f^yz ^f<sf + &c.) 

 V' = - ^sina(\//i + -^iz - xp.^y + ■ip^z'^ - ^\^^yz + i//,^'^ + &c.) 

 W' = «sina(^Ji + (poz - (pay + cp^z^ - (p(,yz + cp^y"^ + &€.) 

 U_|_U', V + V, W-f W, being the components of the whole 

 disturbance. 



