124 Miss A. Everett on a Projective Theorem 



Let K be the pole of the great circle ABM, L the point o£ 

 intersection of the great circles ABM and ODN, and co the 

 angle between them. From K through N, C, D, draw great 

 circles meeting AB at N', C, D' respectively. 



Since the incident ray, refracted ray, and normal at a 

 point are coplanar, the points ABM lie on a great circle, 

 and so do D IN'. 



In the case of the tangential projection, the arcs AC and 

 BD need not be assumed small, but the angles MN between 

 the normals, and CO' between the second ray and the chief 

 plane of incidence, are assumed small of first order. NN' 

 will also be small, 



since 



cosMN = cosNN'.cosMN', .-. NN'<MN. 



DD' may be shown to be of the same order as CO' as follows. 

 From N draw a great circle perpendicular to the great 

 circle which bisects the angle CKD; let it cut KC at Y, and 

 KD at Z ; then the angles at Y and Zare equal, and C'Y = D'Z, 

 hence sin CY : sin DZ : : sin CN : sin DN : : /u': ^ hence CY 

 and DZ are of the same order, and therefore the wholes CC, 

 DD', are of the same order. 



No 



w 



sinNC sin KG sin ND sin KD 



sinNKC sinN ' sinNKD sinN ' 



sin NO sin NK D _ sin KC 

 sinND' sinNKC ~ sin KD ' 



/S s]nN03'_cosOCr__ 

 /i ' sin N'C ~ cos DD' 3 



/*'sinN'D' = ^smN'C', 



and thus the third condition is fulfilled. The projections are 

 conjugate. The positions of A, B on the great circle LAB 

 do not enter into the matter at all. 



The result might also be deduced from the following con- 

 siderations. If the angle of incidence CN is small, so is 

 DN (by the law of sines); the sines of all the small arcs 

 may be taken equal to the arcs, and CC, DD', NN' may be 

 regarded as parallel straight lines dividing two other straight 

 lines NO, N'C, proportionally. 



If ON is finite, a) must be small. The equations 



(i.) sin NN' = sin LN . sin w, (ii.) sin 00' = sin LO . sin co, 

 show that if LN is finite, co must be of first order whatever 



