of Lippicli s in Geometrical Optics. 127 



that the planes of incidence are nearly parallel. Suppose 

 one ray to be incident normally, and the other nearly so ; 

 then the plane of incidence of the nearly normal ray is fixed, 

 while that of the normal ray may have any inclination to a 

 given normal plane. If, however, both rays are incident at 

 90°, and the normals coincide, then « = the angle between 

 the rays, which is small by hypothesis. An idea of the 

 connexion between co and angle of incidence may be gained 

 by supposing the great circles CN, AM, capable of rotating 

 about fixed diameters through N, M, and CA to be a sliding 

 bar with its ends moving on the circles. The remarks above 

 on co at the close of the discussion of the tangential pro- 

 jection apply in this case also, NN', CC (fig. 4), being less 

 than MN, AC, and therefore small. It is seen that for 

 finite angle of incidence, co must be small ; but for small angle 

 of incidence, co may be either small or not. 



Note on the equations of the Projection of the Straight Line, 



x — x' ii — y z — z' 



— j — = - ^ = — ^p on the Plane Lx + my + nz -}-p = 0. 



The equations employed were originally obtained as 

 follows : — 



Let (x ;/ , y" , z") be & second point on the line, at distance r 

 from (x, if, z'). 



Let-p', p" be the perpendiculars from these points on the 

 plane; (£', t) , £'), (%" > v"i K") the co-ordinates of their feet, 

 then 



x" — <r / = Lr, &c. 



p' =p — lx' — my' — nz f , 



g = as > + lp\ &c. 



%"=:X" + Ip", &C. 



The equations of the required projection are 



«—$ _ y-v __ 3—P 



and by substitution 



£'-? = x''-x'-lr(Ll+Mm + lSn) = r(L-lcos»j. 



(If, as suggested below, x", y'\ z" had been taken as the 

 point where the line meets the plane, we should have had 



£' = x", 2 J = rcos0, 



and |" - f = x" - x'- lp' = hr - Ir cos 6.) 



