462 Mr. Sylvester's Analytical Development 



paratory to obtaining the wave surface is found in prop. 6 

 by common algebra, without any use of the properties of 

 maxima and minima, and various other curious relations are 

 discussed. 



Without the most careful attention to preserve pure sym- 

 metry, the expressions could never have been reduced to their 

 present simple forms. 



Analytical reduction of FresneV s Optical Theory qfC?ystals. 



Index of Contents : 



In proposition l,a plane front within a crystal being given, 

 the two lines of vibration are investigated. 



In proposition 2 it is shown that the product of the cosines 

 of the inclinations of one of the axes of elasticity to the two 

 lines of vibration, is to the same for either other axis of ela- 

 sticity in a constant ratio for the same crystal ; and the two 

 lines of vibration are proved to be perpendicular to each 

 other. 



In proposition 3, a line of vibration being given, the front 

 to which it belongs is determined ; and it is proved that there 

 is only one such, and consequently any line of vibration has 

 but one other line conjugate to it. 



In proposition 4, certain relations are instituted between 

 the positions of, and velocities due to, conjugate lines. 



In proposition 5, the z_es made by the front with the planes 

 of elasticity are found in terms of the velocities only. 



In proposition 6, the above is reversed. 



In proposition 7, the position of the planes in which the two 

 velocities are equal (viz. the optic planes) is determined. 



In proposition 8, the position of a front in respect to the 

 optic axes is expressed in terms of the velocities. 



In proposition 9, the problem is reversed, and it is shown 

 that ifu v„ be the two normal velocities with which any front 

 can move perpendicular to itself, and i, \ H the < es which it 

 makes with the optic planes, 



then v* as a 9 (sin '-^-— -* \ + c* /cos /j — ) 



»,;= .(si„^)\ c s(cos'4-'y. 



In the 10th the <es made by a line of vibration with the 

 axes of elasticity is expressed in terms of the two velocities of 

 the front to which it belongs. 



In the 1 1th proposition the velocity due to any line of vibra- 



