Crystalline Reflexion and Refraction. 153 



the semiaxes the angles a, /3, 7, and the other the angles 

 a', j3', 7', such as to satisfy the condition 



cos a cos a' cos J3 cos |3' cos 7 cos 7' , . 



~~^~ ~W~ ~f~ ' 



these diameters will be the axes of the ellipse in which their 

 plane intersects the ellipsoid. 



For, the above condition expresses that either diameter is 

 parallel to the tangent plane at the extremity of the other ; 

 they are therefore conjugate diameters of the elliptic section ; 

 and hence, as they are at right angles to each other, they must 

 be its axes. 



If the semiaxes of the ellipsoid be represented by -, y, -, 



Qi C 



the equation of condition will become 



a z cos a cos a' + b* cos j3 cos ]3' + c 2 cos 7 cos 7' = 0. (G') 



Lemma TV. Let s, s' be the lengths of perpendiculars let 

 fall from the centre of an ellipsoid upon any two tangent 

 planes, and r, r the lengths of radii drawn to the respective 

 points of contact. Then putting w for the angle between the 

 directions of r and s', and a/ for the angle between the directions 

 of / and s, we shall have 



rs cos w = rs' cos to". 



For if the semiaxes of the ellipsoid, having their lengths 

 denoted by , b, c, make with the direction of s the angles 

 a, /3, 7, and with that of s' the angles a', /3', 7'; with the 

 direction of r the angles o , /3 , 70, and with that of / the 

 angles aj, /3i, 71, there will exist the relations 



a? cos o = rs cos a , b 2 cos j3 = rs cos /3 , c 2 cos 7 = rs cos 70, 

 a 2 cos a' = rs' cos ai, 6 2 cos j3' = r's' cos /3i, c 2 cos 7' = rV cos 71, 

 by one set of which the quantity 



2 co r a cos a' + b* cos /3 cos /3' + c 2 cos 7 cos 7' 



