1872.] 



O/i Crystals in Circularly Polarized Light. 



333 



use of, to serve as the origin and insertion of the supplemental muscle. 

 The Table just given shows that in the employment of these supplemental 

 muscles they are always so arranged as to work to the maximum advantage. 



Thus the action of skew muscles, which, on a hasty examination, would 

 be pronounced to he imperfections, furnishes another proof of my postulate. 



Postulate. — The muscles, bones, and joints of all animals are so related 

 geometrically to each other as to produce in every case the maximum amount 

 of work. 



IV. "On the Rings produced by Crystals when submitted to Cir- 

 cularly Polarized Light." By William Spottiswoode, M.A., 

 Treas. U.S. Received April 24, 1872. 



The general optical arrangements here used are known. Particular cases 

 of the phenomena resulting from it have been described by Fresnel and 

 by Airy ; and more have doubtless been observed by others. The main 

 part of the apparatus consists, so far as polarization is concerned, of the 

 ordinary polarizer P, analyzer A, and crystal plate to be examined, C. 

 To this are added two quarter-undulation plates of mica, Q, Q 1} one of 

 which, Q, is placed below and the other, Q x , above the crystal C. Let i, a, 

 b,j be the angles between the principal sections of P, Q, C, Q lf A, taken two 

 and two together in the order written, all the angles being considered to be 

 of the same sign when measured in the same direction — say, positive with 

 that of a clock-hand. Then, if 6 be the retardation produced in any ray, 

 whose wave-length is X, by the crystal C, the intensity of the ordinary 

 image at any point is given (Yerdet, 'Lecons d'Optique Physique/ tomeii. 

 p. 201 *) by the formula 



F=cos 2 (/— i) cos 2 (a + b) + sin 2 O'+O sm2 ( a + ^) 

 + (cos 2i sin 2a sin 2b cos 2j— sin 2i sin 2j) sin 2 - 



+ (cos 2i sin 2a sin 2J+ sin 2i sin 2b cos 2j) sin - cos^. 

 Of this general case four particular instances have been studied, viz. 



«= 6 = 45°, whence I 2 = shr 



a 



= — i=45 c , whence I 2 = cos' : 



(no 



(i) 



% sa ^'=45°, whence I 2 = cos' 



A. 



(in.) 



i = —J =4 5°, whence I a =>sin 2 -. 



• (IV.) 



* Prom which work the greater part of the discussion of the cases L, H., III., IV. 

 has been taken. 



VOL. XX. 2 B 



