99 



eliminating da, d/3, dy, by these equations, from those which are obtained by differentiating the 

 formulae already found, 



«+«' + p(y+y') = o, /i+0 + q{Y+y) = o, 



we get the two following equations, 



0=5.^ (y-f y^^/J + rfa'-f ;;.C?y' \ -\-{a + Yp]{adx +^dj/ -i-ydz) — (dx-^-pdz) 



= {. I (y + V)dq + r//3' + q.dy' | + (/3 + Yq){adx + /3rfy + ydz) — {dy + qdz) (T) 



which give by elimination of f , the following general equation for the lines of reflexion, 



(y -\- yyiq + d/i' + q.dy' __ (/3 + yq)(»dx +^d i/ + ydz) —(dx -\-pd z) .y. 



(y\-y>)dp ^.dei' -{-p-dy' (» -\- yp){itdx -{- lidy-\-ydz) — (dy-\-qdz) * ^ ' 



[34]. Suppose, to give an example, that the incident rays are parallel, and that the axes of 

 coordinates are chosen as in [30.], the normal at some given point of incidence for the axis of 

 (z), and the tangents to the lines of curvature for the axes of {x) and (y) ; our general formula 

 (U) will then become 



'- t.dy _^.(adx-\-fidy) — dy 



r.dx cc.(xdx -^^dy) — dx ' 

 that is 



a/3.it.dy^ — r.dx'') — I (ii^ + y')t — {»''-iy')r \ dx.dy = 0. (V) 



We shall see, in the next section, that the two directions determined by this formula, are the 

 directions of osculation of the greatest and least paraboloids, which, having their axis parallel to 

 the incident rays, osculate to the mirror at the point of incidence ; in the mean time we may 

 remark, that if the plane of incidence coincides with either the plane of the greatest or tlie least 

 osculating circle to the mirror, or if the point of incidence be a point of spheric curvature, one 

 of the two directions of the lines of reflexion is contained in the plane of incidence, while the 

 other is perpendicular to that plane; and it is easy to prove, by means of the formula (V), that 

 these are the only cases in which the lines of reflexion are perpendicular to one another, the in- 

 cident rays being parallel. 



[35.] The formulae (T) determine not only, as we have seen, the lines of reflexion, but also 

 the two focal distances, and therefore the caustic surfaces. For as, by elimination of (5), they 

 conduct to the differential equation of the lines of reflexion, so by elimination of the differentials 

 they conduct to a quadratic equation in (5), which is equivalent to the formula (R), and which 

 determines the two focal distances. As an example of this, let us take the following general 

 problem, to find the caustic surfaces and lines of reflexion of a mirror, when the incident rays 

 diverge from a given luminous point X', Y', Z'. We have here 



