374 PROFESSOR CHALLIS, ON A THEORY OF THE POLARIZATION OF LIGHT 



0-2 shew that at small distances from the axis of z the motion in one polarized ray is parallel 

 to the plane zy, and in the other parallel to the plane z.r. They may be said to be polarized in 

 these planes. 



The foreo-oing reasoning proves that a ray of common light is divisible into two rays polar- 

 ized in planes at right angles to each other, and that these rays are necessarily equal. We have 

 next to shew that they are each of half the intensity of the original ray. Since 



1," + v' = (ur + ij,"') + (m/ + «/)• 



Hence the square of the velocity at any point of the undivided ray is equal to the sum of the 

 squares of the velocities at the corresponding points of the polarized rays. This is true of the 

 velocities in any tranverse section, and therefore true of the maximum transverse velocities. 

 Measuring, therefore, the intensity of a ray by the sum of the squares of the maximum trans- 

 verse velocities, it follows that the sum of the inten.sities of the polarized rays is equal to the 

 intensity of the undivided ray, and, their intensities being equal, that each is of half the intensity 

 of the undivided ray. This is conformable with experience. 



Let us now proceed to estimate the intensity of a ray compounded of two rays polarized in 

 opposite planes, hut not in the same phase. As above we shall consider the intensity to depend 

 entirely on the transverse velocity. In general for any ray not compounded, 



df df ^ m\ 27r , . 



11 = d) ^- , J' = d) — ^, and (b = — cos — (a i — x + c). 

 ^ d.v ^ dy ^ Stt X 



11 

 Now since the ratio - is a function of x and y independent of ;r and /, the direction of the 



?' 



transverse velocity is independent of the phase of the direct velocity. Hence the transverse 

 velocities in two rays polarized in opposite planes, which by hypothesis are at right angles to 

 each other when the rays have the same phase, will be at right angles to each other whatever 

 be the difference of phase. Let therefore for one ray 



df. df, , df, df, 



M, = (h, -^ , V, = (h, -^ ; and for the other, u, — <h, -7— , v., = (h.. -^— . 

 "^ d.T ^ dy ' ^ dx ^ dy 



Tlien since they are polarized in opposite planes, — = , or tt^n., + !),t^ = 0. Consequently, 



df, df df, df. , , 



/^•/^ + T^-/- = (2.3). 



ax dx dy dy 



