i1 



is most complete, and the two reflectors or refractors osculate to each 

 other in all directions. 



On Foci hy Projection, aiid Virtual Foci. 



17. Another kind of focus, of which the' law is similar, though 

 not the same, may be deduced in the following manner. If we con- 

 ceive a plane passing through a given ray of a given optical system, 

 and through a point infinitely near to this given ray ; the ray which 

 passes through the near point may be projected on the plane, and the 

 intersection of its projection with the given ray may be called a 

 focus by projection. Suppose, to simplify the first calculations, that 

 the given ray is the axis of z, and that the infinitely near point is con- 

 tained in the plane oi x y, its coordinates in this plane being denoted 

 by Ix, hy, and the cosines of the angles which the near ray makes 

 with the axes of x and y, being ha, 3/3 : then, if we denote the gene- 

 ral coordinates of this near ray by x„y„ z„, its equations may be thus 



written, : t-ii t\h ,''j'. 



x„ = Sx + z„S«, y„ = Sy + 2„3^, (/f<") 



and the connexions between h<.v, hy, ha, 3/3, will l>e expressed by the 

 two following conditions : 



> (I'") 



which are obtained by differentiating (C) and making hz = 0,hyzz 0. 

 The equation of the plane on which the near ray (H'") is to be 

 projected, may be put under the form 



