of Fresnel's Optical Theorif of Crystals. 75 



Let O be the centre of the wave-surface O A, O B the two 

 prime radii, O P any radius vector. Let O P = u, P O A = <^, 

 P O B = i^p and let the inclination of the planes P O A, 

 POB = /x 



, ,(-i"^)' (-^)' 



then -5- == 



r^ a* c 



(taking only the positive sign for the sake of brevity.) 



Let O Q, O R be the two adjacent radii vectores, so as- 

 sumed that 



QOA = POA QOB = POB + 8*/, 

 ROB=POB ROA=POA-f8i; 



and let p^ q, r, a, b be the projections of 

 P, Q, R, A, B on a sphere of which O is the 

 centre, then it is clear that 



qpa=z9(f rpbzrz 90° 



draw q m perpendicular to pb , then p m = li^ 



pm 



and .*. pq = . = 



^ ^ smpqm 



In like manner pr = 



Now the angle Q P O 



^ -1 r.POQ ^ 



=-(^-^)(-'"^')(~H^) 



dr 



rd 



T* /I 1 \ 



.•.cot.QPO = ~ . y^- -^)sin(»;-f *).sin|ic 



In like manner 

 cot. RPO = — . y-^ --^jsin(i^ + 0-sin^ 



.-. Q P O = R P O. 



L2 



