126 Miss A. Everett on a Projective Theorem 



Projecting on the sphere (fig. 6), and letting ABMNCDKL 

 have the same meanings as before, the great circles KA, KB 

 represent the sagittal sections, and KM represents the second 

 principal section at 0, 



From N draw a great circle NN" at right angles to KM, 



„ C JJ J5 5? CC ,, „ ,, KA, 



„ D „ „ „ OD" „ „ „ KB. 



In this case MN and AC are assumed small of first order. 

 Hence MN", AC", are also small, being less than MN, AC. 



MK is a principal section at M, and therefore at N" near 

 M, hence NN" at right angles to MN"K is the other principal 

 section at N", and the line represented by N", being the line 

 of intersection of two principal sections, is a normal to the 

 refracting surface. 



If CN is small of first order, so is UN, and as before all 

 the small arcs in the region NCC'AM may be taken as 

 straight lines, and equal to their sines. Hence, since 

 NC : ND : : MA : MB, it is easily seen that the points 

 N", C", D" lie in a straight line, and therefore the lines they 

 represent are parallel to a plane, and having been shown to be 

 concurrent, must be coplanar. Also N"C": N"D": : jjJ \ /jl. 

 Hence, if the angle of incidence is of the first order, the 

 sagittal projections are conjugate, neglecting second order 

 quantities, and obviously the smaller the angle of incidence 

 the closer the approximation. 



If the angle of incidence is not small, then by neglecting 

 small quantities of the first order, N may be taken as co- 

 inciding with M, C with A, and therefore I) with B. Hence 

 the theorem holds with that proviso. 



Angle co between the planes of incidence. 



Because two near incident rays are nearly parallel, and the 

 normals also nearly parallel, it does not necessarily follow 



