in Singly and Doubly Refracting Media. 513 



cular to OS, the question is, the angles it makes with the axes. 

 If now we join the points X, X, Z, also R, X, S, as well as 

 R, X ; R, Y ; and X, Y, at the unit of distance from by 

 arcs of a circle, there will result the two spherical triangles, 

 RXX and RXY. 

 In the first we have 



cos RX = cos RX cos XX + sin RX sin XX cos RXX, 



or, since RS = 90°, XS = A, and the surface-angle RXX = 0, 

 i. e. equal to the azimuthal angle of the vibration-plane ROS 

 and the plane of incidence XOZ, 



cos RX= cos (90°- A) cos (90° - r) 



+ sin (90° -A) sin (90°-?') cos 6 ; 



cos U = sin A sin r + cos A cos r cos 6. 



The triangle RXY has the side YX=90°, and the surface- 

 angle YXR, which is distant the azimuth = 90° from YXX. 

 Hence we find 



cos V = cos RY = sin RX cos RXY 

 = cos A sin 6. 



If, lastly, we draw, in the vibration-plane SXR, and per- 

 pendicular to OX, the vibration-direction 02ft, and join it by 

 the arc 2ftZ with Z, in the resulting triangle 2&ZX the side 

 2&X = 90°; consequently 



cos M = cos 3&Z = sin 2&X cos 2&XZ 



= — sin r cos 6. 



There still remains the measurement of the perpendicular h. 

 Again, let the plane of the paper be the plane of incidence, 

 OX the line in which it intersects the dividing surface, and 

 OZ perpendicular to the incidence. The wave-normal OX is 

 cut in the point A, and the X-axis in the point D, by the cor- 

 responding wave-plane ; DA is then its projection on the 

 plane of incidence. In the wave-plane, therefore in general 

 above or below the plane of the paper, lies its point of contact 

 B with the wave-surface, which determines both the direction 

 OS and that of the vibration AB. 



Xow, instead of letting fall directly from B a perpendicular 

 upon the dividing surface, let us first drop from B upon DA 

 produced the perpendicular BC meeting it in C, and from C 

 the further perpendicular CE upon the axis OX, which it ac- 

 cordingly meets in E. Then CE is also the prolongation of 

 the perpenicular h. We have now in succession : — 



Phil Mag. S. 5. Xo. 14. Suppl, Vol. 2. 2 L 



