On the Laws of the Double Refraction of Quartz. 73 



mark suggested* to me the idea of adding, to the equations of the 

 common theory, terms containing the third differential coefficients 

 of the displacements ; for it was evident that such additional 

 terms would give, in the value of s 2 , a part inversely proportional 

 to /. It was also evident that the third differential coefficient 

 of should be combined with the second differential coefficients 

 of j, and the third of rj with the second of , in order that, after 

 substitutions such as we have indicated in deducing formulae (5) 

 and (6), the sines or cosines might disappear by division, and 

 that thus the value of s 2 might be independent of the time, as 

 it ought to be. This kind of reasoning led me to assume the 

 equations 



for the case of a ray passing along the axis of quartz ; and then, 

 substituting in these equations the values of the differential co- 

 efficients obtained by differentiating the formulae 



= p cos -8t-z , rj = p sn -=- 

 if ; 



which express a circular vibration (from right to left, or from 

 left to right, according to the sign of the second p) , the result 

 was 



S * = a' + ^ C 



I 



from (19), and 



S > = ' D 



* " The singular relation between the interval of retardation [S] and the length 

 of the wave [7] seems to afford the only clue to the unravelling of this difficulty." 

 " Eeport on Physical Optics," by Professor Lloyd (" Fourth Report of the British 

 Association," p. 409). It was in reading this Report that Fresnel's remark, about 

 the relation between 8 and I, first came to my knowledge. 



