100 



f' being the distance of the luminous point from the mirror; 



«?j'= — {a'dx + fi'dl/ + y'dz) 



—^.d«! = dx+»'.d^', —il.d^'zzdy + ^'.d^', —(.d^=.dzJfy'.d(, 

 and because 



«'+y';j=_(« + ^), ^' 4- y'y = — (/3 + yy) 

 a'cfx + /3'rf^ 4- yrf« = — («rfx + /3rf^ + yrfz), 



the equations (T) become 



(v + yO- «5p = (- + y) ^ dx-\-'pdz—{» ■\-vp){»dx + ^dy-\-ydz) \ 



eliminating j, we find, for the lines of reflexion, 



dq. \ dx-\-'pdz—{»-\-'/p){cids-\-^dy-\-^dz) j = 



dp. >^dy^rqdz~-(_^^ryq){*dx-\-^dy\-ydz) \ , (W) 



and eliminating the differentials, we find, for the focal distances, 



X { (l+?'-(^+y?)^) (j + -^)-(y+y')- * } 



= |(/'?-{«+y?')(^+y?))(-+ v)— (y+V).*}'- (^) 



We may remark, that since ({') has disappeared from the equation (W) of the lines of re- 

 flexion, the direction of those lines at any given point upon the mirror depends only on the di- 

 rection of the incident ray, and not on the distance of the luminous point ; we see also, from the 

 form of the equation (X), that the harmonic mean between that distance (§') and either of the 

 two focal distances ({), does not depend on (j') : so that if the luminous point were to move along 

 the incident ray, the two foci of the reflected ray would indeed change position, but the line 

 joining each to the luminous point, would constantly pass through the same fixed point upon the 

 normal. 



