INTERFERENCES OF CIRCULARLY POLARIZED RAYS. 203 



velocities, there mast be certain tliicknesscs of tlie crystal, wliicli will make tlie 

 difference of their paths equal to half an undulation, or to an odd midtiplc of 

 half an undulation. It might be supposed, therefore, that in such cases inter- 

 ference would occur, so that the crystal should naturally exhibit colors. The 

 fact is not so ; and if we consider the conditions Ave shall discover without diffi- 

 culty the reasons why it is not. If, in Fig\ 52, the two components q and r' of 

 one of the circularly polarized rays be supposed to advance or gain upon q' and 

 r, the distance between q and q' will diminish until q passes q', and the distance 

 between r and r' will constantly increase. If c represent the amount of advance, 

 the distance of r' from the plane of qr in the figure will be c-f-^A. ^^<^ the disr 

 tance of q from the same plane will be c only. Now, since q and q' are equal, 

 the resultant to Avhich they are equivalent will fall half Avay between them, 

 (page 167.) The same is true in regard to the resultant of r and r'. But the 

 point half way between q and q' will be situated at a distance frora the plane 

 qr which is the mean of the distances of q and q' ; thus — 



Distance of q = c ) Mean — -ifc-l-i;)— ic 4-1/1 

 Distance oW = ll | Mean— ^(c + ^/j — vc-f- ^x. 



In like manner — 



Distance of r = \ , . _ w„ , i ,n _ i„ , , , 



Distance of / = c + -i;. } Mean^-.l^+^/j — ,c-f ^X. 



That is to say, the resrjtants of the components in each plane always coincid*^ 

 in position. We have next to consider their values. From the statements 

 above it appears that the distance hctioccn r and ?•' is the entire distance of r* 

 from the plane qr — that is, ^c+^A. And the distance between q and (f is the 

 difference of the distances of q and q^ severally from the same plane qr — that is, 



In the general equation for the resultant of two waves whose molecular move- 

 ments are in the same plane, (equation [3,] ) we must accordingly introduce th>j 

 followinsx values of 6' : 



For q and (f , = 2~(^-—\\. 

 For r and /, 0^2-(^- + \\. 



Then putting p for the first resultant, and p' for the second, and remembering 

 that q^iq':z^r--^^r' , the equations become, 



p'=q^ + q-'-^2qq'i:0^:>r.(^: — ^^^q'J^q''~ + 2qq'^m27z'^~^^^ 



y-^?-- + r" + 2//cos2-( ^. + M = 7-2 + r'- — 2;T'sin2-^'==2/-Y^ 



Whence ^r4-7'/"-~2'y^+2/'"=:'0« /rt///. And the intonsity of the light is invi- 

 riable. 



By considering tlic foregoing values of p^ and ^/'^, however, it will be seen 

 that they are severally variable, though their variations are always compen- 

 satory. If c be any number of half undulations — 



(^ 

 Sin2- = 0: and »^ = //^. 



But if c be an odd number of quarter undulations, 



Sin2-- = 1 or — ] ; and either ^^=0, ory^ = 0. 



Both 2^^ and iP, therofore, pass through a succession of maxima and minima, 

 the increments of the one corresponding always in value to the simultaneous 



