124 



R'. COS. »« + R" sin. *« ' R' cos. "• »> + R" sin. *"« 



;R'cos. «« + iJ''sin.V ' ^ ' 



, i.(fl' — R"). sin- «. COS. a 



R, R" being the two radii of curvature of the caustic surface. It appears from these formulae 

 (T"), that when the ray touches]either line of curvature uppn the caustic surface, (which is always 

 the case when the reflected system consists of rays, which after issuing from a luminous point, 

 have been reflected by any combination of mirrors of revolution, that have a common axis pass- 

 ing through the luminous point), then B vanishes, and the area (Q") of the little hour-glass 

 curve is equal to nothing. In fact, in this case, that curve changes shape, and becomes 

 confounded with a little parabolic arc, which has for equation 2jiy=/4x'»4- C(j^«'' — x'^), and 

 which is comprised between the limits x' = ■±lU; this parabolic arc is crossed at its extremities 

 by the parabola (R")> of which the equation becomes 2t V = An!- : and the whole space in- 

 cluded between these two parabolas, that is the whole space over which the near rays are dif- 

 fused, has for expression, 



2' = -| i..C.f. (U") 



[62.] As a third application, let us consider the case of aberrations from a principal focus. 

 In this case we have i = 0, and the expressions for 2,2', vanish; we must therefore have re- 

 course to new calculations, and introduce terms of the second order, in the expression of of, as 

 well as in that oi y'. We find 



expressions which may be thus written 



(V") 



A, B, C, having different meanings here, from what they had in the preceding paragraphs. 

 And if we eliminate «,, /3„ between these equations, by means of the relation 



»," + A» = f, 



which expresses that the near rays make with the given ray an angle =^; we find, for the 

 curve of aberration, that is, for the locus of the points in which those rays cross the perpendi- 

 cular plane drawn through the principal focus, the following equation, 



