Crystalline Reflexion and Refraction. 165 



cone, which lie in the plane of xz, is given by the angles and 

 0' which the direction of the right line OTT' makes with the 

 nodal diameters ; because the angles which any side of a cone 

 makes with its focal lines have a constant sum, or a constant 

 difference, according to the way in which they are reckoned. 

 But if the angles and 0' be reckoned (as they may be) so that 

 their sum shall be equal to the angle contained by the two sides 

 of the cone B which are in the plane of xz, their difference will 

 be equal to the angle contained by the two sides of the cone B' 

 which are in the same plane ; the contained angle, in each case, 

 being that which is bisected by the axis of x. Therefore, the 

 lengths OTand OT', which we denote by r and r', are equal to 

 two radii of the ellipse whose equation is 



- + -r = 1, 



a 2 c 2 



these radii making with the axis of z the angles ! (0 + 0') and 

 ! (0 - 0') respectively. Hence 



_!_ = . sin 2 ! (0 + 0') cos 2 ! (0 + 0') ' , / 1 



(13) 

 _! = sin 2 ! (0-0') + cos 2 !(0-0')_ , /I ^ 



r' 2 a 2 c 2 ~ 2 Va 2 



These formulae give the two velocities of propagation along a 

 ray which makes the angles 0, 0' with the nodal diameters. 

 Subtracting them, we have 



n 9 sin ff. (14) 



All the preceding equations, relative to the wave-surface, 



