90 Relative Retardation between Components of a Stream of Lwht. 



and the difference of the positive roots being 2/> + a 7, we have, to 

 the same degree of approximation, 



su 



v/O a -6 2 )(Zr-c-) sin*', // 4 -aV shri 



-p- - -l-i.-ji-cos* 



1 (a 2 -c 2 ) 2 . ., ' 



a. . ., 



' S1 



*> T * F(a' - 6*) 



x (2(a 2 -6 8 ) (6 3 -c 3 ) (/> 4 





6. The proposition on which the above investigation depends was 

 first suggested to me by an analogous theorem given by MeCullagh,* 

 in connection with the surface of wave-slowness, f or, as he terms it", 

 the surface of refraction or index surface; in fact, the one may be 

 deduced from the other by reciprocating with respect to a sphere of 

 unit radius concentric with the surfaces. 



I have since found that Zech^ has employed the same principle 

 for the determination of the rings of biaxial crystals, but his method 

 of dealing with the biquadratic equation is essentially different from 

 that given above, and leads only to the determination of the terms of 

 the second order. 



My thanks are also due to Mr. J. L. S. Hatton for some useful 

 suggestions that led me to the adoption of the above methods of 

 approximation . 



* ' Collected Works,' p. 46. 



t The first pedal of the wave-surface is sometimes erroneously called the surface 

 of wave-slowness j but, as Sir William Hamilton calls the inverse of the wave- 

 velocity the wave-slowness, the inverse of this surface, or the polar reciprocal of 

 the wave-surface, is properly the surface of wave-slowness. That this was the 

 name given to the polar reciprocal of the wave-surface by Sir William Hamilton 

 appears from Lloyd's " Eeport on Physical Optics " (' Collected Works,' p. 122), and 

 from MeCullagh (' Collected Works,' p. 96), though in his papers he calls it the 

 surface of components of normal slowness. 



J ' Fogg. Ann.,' vol. 97, j>. 129 ; vol. 102, p. 354. 



