86 



[10.] Then, in general, if it be required to find a mirror which shall reflect to a given focus the 

 rays of a given system, we must try whether the rays of that system are cut perpendicularly by 

 any series of surfaces ; for unless this condition be satisfied, the problem is impossible. When 

 we have found a surface cutting the incident rays perpendicularly, we have only to take upon 

 each of those rays a point such that the sum or difference of its distances, from the perpendicular 

 surface and from the given focus, may be equal to any constant quantity ; the locus of the points 

 thus determined will be a focal mirror, possessing the property required. Or, which comes to the 

 same thing, we may make an ellipsoid or hyperboloid of revolution, having a constant axis, but a 

 variable excentricity, move in such a manner that one focus may traverse in all directions the 

 surface that cuts the incident rays perpendicularly, while the other focus remains fixed at the 

 point through which all the reflected rays are to pass ; the surface that envelopes the ellipsoid or 

 hyperboloid, in all its different positions, will be the mirror required. And to determine whether 

 the reflected rays converge to the given focus, or diverge from it, it is only necessary to deter- 

 mine the sign of the distance 5', which is positive in the first case, and negative in the second. 



III. Surfaces of constant action. 



[11]. We have seen, in the preceding section, that if it be possible to find a mirror, which 

 shall reflect to a given focus the rays of a given system, those rays must be perpendicular to a 

 series of surfaces ; and that the whole bent path traversed by the light, from any one of these 

 perpendicular surfaces to the mirror, and from the mirror to the focus, is a constant quantity, the 

 same for all the rays. Hence it follows, reciprocally, that when rays issuing from a luminous 

 point have been reflected at a mirror, the rays of the reflected system are cut perpendicularly by 

 by a series of surfaces ; and that these surfaces may be determined, by taking upon every re- 

 flected ray a point such that the whole bent path from the luminous point to it, may be equal to 

 any constant quantity. I am going to shew, in general, that when rays issuing from a lumi- 

 nous point, or from a perpendicular surface, have been any number of times reflected, by any 

 combination of mirrors, the rays of the final system are cut perpendicularly by a series of sur- 

 faces, possessing this remarkable property, that the whole polygon path traversed by the light, in 

 arriving at any one of them, is of a constant length, the same for all the rays. 



[12.] To prove this theorem I observe, that if upon every ray of the final system we take a 

 point, such that the whole polygon path to it, from the luminous point or perpendicular surface, 

 may be equal to any constant quantity, the locus of the points thus determined will satisfy the 

 differential equation 



,_ ,4.« + A.,y + A.,z=0, (K, 



'X y, Z, being the coordinates of the point, and ^ the last side of the polygon ; because by hy- 

 pothesis the variation of the whole path is nothing, and also that part which arises from the va- 



