143 



be satisfied by a proper direction of the positive portions of the axes of y' and x' ; our ex- 

 pressions for a, h, become 



a = «. P-*. r. sec. »'. sin. (u — v')> 6= r±: \/(8.»". sin. d), (L'"') 



and we find the following equations for the boundary of the space SC"), 



1st. P. a =.6'^. sec. v'. cosec. c ,. sin. (u, — v') "J 



?d. P.o = 6''. sec. v'. cosec. «,. sin. (v^ — «') > (M"") 



3d. P.a = 6^ q: tan. v'. ^(i\r^~ b*). J 



The two first of these equations represent parabolic arcs, having their common vertex at the 

 origin of (n) and (b), that is at the point where the given ray meets the mirror, and having their 

 common axis coincident with the axis of (x) or of (a), and therefore parallel to the tangent of 

 the curve in which the caustic surface is cut by the plane of aberration. It is, then, in the 

 points of these little parabolic arcs, that the rays which pass through the bounding radii of the 

 little circular sector, are intersected by the perpendicular plane at the mirror; and from the 

 manner in which their parameters depend on the inclination of those bounding radii to the tan- 

 gent of the caustic surface, it is evident that any intermediate radius of the sector has an inter- 

 mediate parabola corresponding. The ends of these little parabolic arcs are contained on two 

 equal and opposite portions of a curve of the fourth degree represented by the third of the three 

 equations (M"") ; it is then in these two opposite portions of this third curve, that the rays which 

 pass through the bounding arc of the sector are crossed by the perpendicular plane at the mirror. 

 With respect to the form of this third curve, considered in its whole extent, it is easy to see that 

 it is in general shaped like a heart, being bisected, first by the axis of (a), which we may call 

 the diameter of the curve, and secondly by a parabola 



b"" = P.a, (N'"') 



which we may call its diametral parabola, and bounded by the four following tangents, 



1st. b=-{-\^t7; 2d. b= — ■/tJ; 3d. n = — i.r.P-K tan. v ; 

 4th. a =■ t.r.P-K sec. v', (O"") 



of which the two first are parallel to the diameter, and the two last perpendicular thereto. We 

 may remark that the diametral parabola, (N""), corresponds to the rays that pass through the 

 axis of y, that is, through the normal to the caustic surface ; and that the two points where it 

 meets the curve, are the points of contact corresponding to the two first of the four tangents 

 (O""). The point of contact corresponding to the third of these tangents, is situated at the ne- 

 gative end of the diameter ; and the fourth touches the curve in two distinct points, whose 

 common distance from the diameter is i = :±z '/{i-r- cos. i/), and which may be called the two 

 summits of the heart. The curve has also another tangent parallel to the axis of (b), which 

 touches it at the point , 



a= + i.r. P-K tan. t/, i =r 0, ( P"") 



VOt. XV. X 



