114 



is greatest, and of which each is the conjugate focus of an ellipsoid of revolution, that has its 

 other focus at the given luminous point, and that has contact of the second order with the given 

 mirror. It is evident that these points of maximum of density are the images of the given lu- 

 minous point, formed by the given mirror; and that in like manner, the image or images of a 

 given point, formed by a given combination of mirrors, are the corresponding points of maximum 

 density, to which the intersection of the last pair of caustic surfaces reduces itself, and which 

 are the foci of focal mirrors that have contact of the second order with the last given mirror. 

 And on similar principles are we to determine the image of a curve or of a surface, formed by 

 any given mirror, or combination of mirrors ; namely, by considering the image of the curve or 

 surface as the locus of the images of its points. 



[52.] Let us apply these principles to the investigation of the image of a planet formed 

 by a curved mirror. The image of the planet's centre is the focus of a paraboloid of revolution, 

 which has its axis pointed to that centre, and which has complete contact of the second order 

 with the mirror. To find this image, together with the corresponding point of contact, or ver- 

 tex on the mirror, we have the equations 



} (B") 



» + *'+p.(Y + y') = 0, /3-}-/3'-f9.(y + y')=0, (A") 



(y + '/)-i<^P — d"^ -i-pdz — («-}- yp){»dx -\- lidy -|- ydz) 

 (y .j. y').^dq ■=. dy -If qdz — (|3-|- yq)[»dx 4- /3rfy -\-ydz) 



a, |3, y, «', /3', y, being, as before, the cosines of the angles which the reflected and incident 

 rays make with the axes of coordinates, and (g) being the focal distance; the formulae (B") are 

 satisfied by every infinitely near point upon the mirror, and therefore are equivalent to three 

 distinct equations, which contain the conditions for the contact of the second order between the 

 paraboloid and the mirror. Differentiating the equations (A"), in order to pass from the centre 

 to the disk of the planet, and eliminating {dp, dq) by means of (B"), we find 



= 5.{rf<«' + prfy')+ dx -I- ^d»-\-p.(dz-\-^dy) — {*-\-yp){»dx-'t-fidy-\-ydz) 

 = g [d^' -4- qdyf) ■\-dy + frf/3 -J- q.(dz-\- jr/y) — {,3 4. yq){»dx-\-fidy -{- ydz) 



that is, 



= ^.(rfa' -\- pdy)-\-da-^pdc — (« + y/))(«afa ■\- ^db -\- ydc) 



= j.(rf^' + qdyf) +db-\ qdc — (/3 ■\-yq){ada -j- lidb ■\- ydc ) 5 (^ ") 



if we put {a, b, c) to represent the coordinates of the image, so that 



a — X := «{, b — y := /3g, c — z := yg. 



Differentiating also the three distinct equations which are included in (B"), and eliminating, we 

 shall get a result of the form 



^.dy' = Ada + Bdb -f- Cdc, (D") 



A, B, C, involving the partial differentials of the mirror, as high as the third order. These 



