109 



(u) being a constant coefficient, 



M = ({, — {j). sin. L. COS. L, (O') 



and (J) being the distance of the point of contact from a certain fixed point upon the ray,' whose 

 distance from the mirror is 



a/ = {, COS. 'Z + {, sin. =L. (P^ 



[^S.] The quantity (u), which thus enters as a constant coefficient into the law of rotation of 

 the tangent plane of an undevelopable pencil, I shall call the coefficient of undevelopahility. 

 In the third part of this essay, I shall treat more fully of its properties, and of those of the fixed 

 point determined by the formula (P) ; in the mean time, I shall observe, that if we cut the con- 

 secutive ray (K') by any plane perpendicular to the given ray, at a distance (S) from this fixed 

 point (P'), the interval between the two rays, corresponding to this distance (3), is 



A = a/(«'' + §=). di, ' (Q') 



{di) being the angle between the rays ; from which it follows, that the fixed point (P') may be 

 called the virtual focus of the given ray, in the given undevelopable pencil, because it is the 

 nearest point to an infinitely near ray of that pencil ; and that the coefficient of undevelopabi- 

 lity (u), is equal to the least distance between the given ray and the consecutive ray, divided by 

 the angle between them. We may also observe, that although a given ray has in general an in- 

 number of undevelopable pencils passing through it, and therefore an infinite number of virtual 

 foci corresponding, yet these virtual foci are all included between the two points where the ray 

 touches the two caustic surfaces, because the expression (P') 



a' = 5i cos. ^Z, + g,. sin.'Z,, 



is always included between the limits {, and j^- And whenever, in this essay, the term /bci of 

 of a ray shall occur, the two points of contact with the caustic surfaces are to be understood 

 except when the contrary is expressed. 



X. On the axes of a reflected system. 



[46.] We have seen that the density of light in a reflected system is greatest at the caustic 

 surfaces ; from which it is natural to infer, that this density is greatest of all at the intersection 

 ■ of those surfaces : a remark which has already been made by Malus, and which will be still 

 farther confirmed, when we come to consider the aberrations." It is important therefore to in- 

 vestigate the nature and position of the intersection of the caustic surfaces. I am going to shew 

 that this intersection is not in general a curve, but reduces itself to a finite number of isolated 

 points, the foci of a finite number of rays, which are intersected in those points by all the rays 

 infinitely near them. For this purpose, I resume the formula (Q) found in Section VII. 



