Transparent Inactive Crystal Plates. 173 



To apply this relation, discovered by Potier, between two 

 pairs of corresponding points on the two ellipsoids (F) and (I), 

 it is necessary to obtain the coordinates of the points. In the 

 stereographic projection (tig. 5), let p and P be the projections 

 of the two corresponding points j» and P, ~Ky' the wave front, 

 x'z' the plane of incidence, N the wave normal, r the angle of 

 inclination of N with z', yjr the azimuth of the plane of polari- 

 zation, and s the angle ^>OP (O being the center of the sphere). 



The coordinates of p are then: 



a;'=0» cos px'= sin iA cos r, 



q x 



y' t = Op cos py' = cos ij/ 1 (2*7) 



ii 



- ^ i 1 ■ 



z , = Op cos pz = — sin \p x sin r x 



"i 



and the coordinates of P are 



ai'^OP cos Pa;' = OP (sin s 1 sin r x — cos s, cos r x sin iAJ 

 y' i = OP cos Py'=z — OP cos s, cos i£ x (28) 



z'^OP cos Yz' =OP(sin s x cos ^4- cos s x sin r t sin i/^) 



Op' q 



But from fig:. 4, OP= 1,^ , = — — — and equations (28) can 



& ' cos POji/ cos s, H K J 



be written: 



^"^q^ty s 1 sin r, — cos r 2 sin i/^) 



y'°.= -?,co8^ _ _ (28a) 



z'" i = q 1 {tys 1 cos j^+sin r t sin t^J 



In tig. 5, P is situated between JM and p ; in case P lies 

 beyond^?, tg s, in (28a) changes sign and becomes negative. 

 If the sign ± be placed before tg s x , therefore, all possible 

 relations are taken into account ; for each particular case, the 

 proper sign must be determined. 



On substituting these coordinate values (27) (28a) in the 

 Potier relation (26) for two sets of corresponding points p x , P l5 

 j? 2 , P, of the waves ¥„¥, whose normals lie in the plane of 

 incidence x'z, we obtain, 



— sin \fi x cos r x sin y^ cos r 2 — sin if> 1 cos r x tg s 2 sinr 2 + 



cos y x cos v„ + sin y x sin r x tg s 2 cos r a + sin if x sin r x sin v 2 sin »•„] 



= — sin y 2 cos r 2 sin ^ cos r x — sin i/> 2 cos r 2 ^ s t sin r 1 ■+- 



cos i// 2 cos v, 4- sin «/> 2 sin r 2 tfy Sj cos r x + sin y 2 sin r 2 sin i/\ sin rj 



This equation may be rearranged to read 



(<?\-?\)[ sin V, sin V, cos (r 1 -r 2 ) + cos V, co* y t ]— . . 



(q\ tg s, sin y^ + q\ tg s„ sin y x ) sin (?*,— r 2 ) = v ' 



Am. Jour. Sci.— Fourth Series, Vol. XXXI, No. 183.— March, 1911. 

 13 



