CHROMATICS — DISCUSSION OF THE FORMULAE. 209 



being squares, liavc tlieir essential the same aa their writton signs ; hut the 

 negative element of this coefficient is the same as the positive achromatic term. 

 Hence the entire coefficient can never be greater than the achromatic term; and 

 can only be equal to it in the single case when cosine or sine (2a — (i)=0. But 



the chromatic term has another factor, sin^^r, which is always less than unity, 



except when 7i is an integral odd number of half undulations. This, therefore, 

 usually still further reduces the value; so that neither of the expressions for 

 intensity can ever become negative. 



Fifth, if /3 remain constant, the value of the chromatic term will vary with a, 

 and may even become zero Avhen m=:/3. The force of the color will therefore 

 undergo corresponding variations ; and all color will disappear in the case just 

 mentioned. The same also will be true when a=r/5-f-90°, /3-}-180°, &c., (fee. 

 But though, in these positions of the lamina white light only is seen, the color 

 reappears for values of a intermediate between /3 and .^-f-90°, /3-)-90° and 

 /34-180'^, &;c. ; and this color is the same as before, since the sigu of the chro- 

 matic term does not change. When a is constant and /? varies, the color in like 

 manner rises and descends in brilliancy, having a minimum =0, at the values 

 P^a, /3=a-|_90° and P=a-\.180° , &c. But as, in passing each of these successive 

 values, the coefficient of the chromatic term, as is evident on inspection, changes 

 its essential sign from positive to negative, or the contrary, the tints observed in 

 the successive quadrants will be complementary to each other. 



Sixth, Avhen a=0°, 90°, ISO'^, &c., the chromatic term disappears for evpry 

 value of /3. In this case the light remains white throughout the entire revolu- 

 tion of the analyzer, and one or the other of the achromatic terms disappears, 

 for the azimuths /3=0°, /5=90°, &c. 



Seventh, if wt- suppose the lamina and th? analyzer both to remain .-stationary 

 while the polarizer revolves, we shall see that the chromatic term changes its 

 sign in the course of every quarter revolution. For example, since tlic change 

 of plane of original polarization affects the azimuths of the lamina and of the 

 analyzer equally, if we suppose a revolution of 90° in the negative direction, 

 the coefficient cos2(2a— ;?)— cos^;? becomes cos\2a-\-]S0°—90^—i3)—cos^^90^ 

 +/5)=cos2(90°+2«— /?)— cos2(9a°+/3)=sin2(2a— ;9)— sin^/?. But, by reference 

 to the two equivalent values of A^, [37. J [38,] we see that cos^(2a — jS) — cos^/?= 

 — (sin^(2a — /5) — sinV) Hence, in the rotation of the plane of polarization 

 through an arc of 90°, the coefficient of the chromatic term passes from positive 

 to negative or the contrary. If we suppose a i-evolution of 180° still in the 

 negative direction, we shall find the sign once more the same as in the original 

 expression. Thus, cos2(2a — fi) — cos^ becomes cos2(2a-f-360° — 180° — ;?) — 

 cos2(180°+,3)=:=cos2(180°+2a— ;5)— cos2(180°+/5)=^cos2(2a— ,5)— co.s2,?. If we 

 suppose the rotation in the opposite direction, the alternations are similar. 



It is, hence, manifest that unless the light employed in these experiments be 

 originally ])olarized, no chromatic phenomena will make their appearance. For 

 unpolarizcd light being made up of successive undulations impartially distributed 

 through all azimuths, those which are embraced within any one quadrant will 

 neutralize the effects of those within the adjacent one, the complementary colors 

 produced by each similarly situated pair succeeding each other with such rapidity 

 as to blend their effects upon the retina. 



Eighth, if wc consider the factor sin^--, we shall see that, when /i=lX, iU. 



f z'., &c., the value of this factor is unity, Avhich is its greatest possible value. 

 The chromatic effect is, therefore, greatest when the retardation of one ray upon 

 the other is an odd numbfr of half undulations. If, in this case, ,9=2a, then 

 A^^V^, and A'2=0. If /3-=2a+90°, A^^O, and A'^^V^. Supposing, there- 



