290 Sir W. R. Hamilton on the Focal Lengths and 



tering a given uniaxal crystal, of which the optical axis co- 

 incides with the axis of revolution and of z. It is required 

 to determine the intersection of the refracted ray with the 

 axis; the distance of the point of incidence from the vertex, 

 or the semiaperture of the crystal, being given. 



2. The law of this extraordinary refraction may (according 

 to the general methods of my ' Theory of Systems of Rays,' 

 published in the Transactions of the Royal Irish Academy,) 

 be thus expressed : 



(<r-«')8£ + ( y -y')8$ = (I.) 



In this formula, £, £ are (in the plane of x z) the rectangu- 

 lar coordinates of incidence, and are connected by the equa- 

 tion of the meridional section of the given refracting surface 

 of revolution, which equation we may suppose to be (at least 

 nearly enough for our present purpose) developed under the 

 form 



? = i?+T 5 * W 



r being the curvature at the vertex, and s being another con- 

 stant, which in the case of a spheric surface is half of the 

 cube of that curvature ; «', y' are the cosines of the inclinations 

 of the incident ray to the positive semiaxes of x t t so that 

 they are connected by the relation 



a' 2 + y' 2 = l ; (3.) 



and a-, u are the components of normal slowness of the extra- 

 ordinary wave within the crystal, so that, if /x be the ordinary 

 and v the extraordinary index, these components are con- 

 nected by the relation 



ju. 2 o- 2 + v 2 u 2 = ju, 2 v 2 (4.) 



And these quantities, a' y' a w, are such, that if x' 2' be the 

 coordinates of any point on the incident ray, we have 



(£-*') $*'+{$- z')Zy' = 0; .... (5.) 

 and if xy be the coordinates of any point on the refracted ray, 

 we have 



(#- £)»<>■ + (*-§) 8 u = (6.) 



3. Making then 



T = £(<r- a ') + ^-y')> (7.) 



we have first the equation 



w=°> ■■■- ;••• (8 ° 



which contains the law of extraordinary refraction, and by 

 which £ can be eliminated from the expression of T, so as to 



