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 TJ, and the third of v\ 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 



n _ 



- = a 2 -j-7 1 + D -5-5 -, (20) 



df* dz 2 dz z 



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 



\2ir . . . ) . (2;r , . . 



=p cos (-)>, n = p sm (st-z) 

 l * ) v l 



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 z + ^ C 

 t 



from (19), and 



s* = a* ~ D 

 i 



* " The singular relation between the interval of retardation [8] and the length 

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

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

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

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



