46 



We may determine the four arbitrary constants of F'j in this equa- 

 tion, by the condition that the focal reflector or refractor shall touch 

 the given reflector or refractor at a given point, and osculate in a 

 given direction. The condition of contact, of the first order, is ex- 

 pressed by the equations 



and may be satisfied by establishing between the three coordinates of 

 the focus the two equations of the ray, and by assigning a proper 

 value to the remaining arbitrary constant ; and the position of the 

 focus upon the given ray, may be determined by the condition of os- 

 culation in the given direction, which is expressed by the equation 



assigning the given ratios to the variations Sx, ly, hz. This equation 

 (G'") being the same with that marked (X") in the foregoing num- 

 ber, we can deduce from it the same consequences ; the osculation 

 thei'efore between the focal surface (E'") and the given reflector or 

 refractor, follows the same law as the osculation between the spheroid of 

 constant action (V") and the given surface (JJ") for which the function 

 V is constant ; in such a manner that the focus of the focal reflector or 

 refractor coincides with the centre of the spheroid, if the point of 

 contact, and the plane of osculation be the same. The distances 

 therefore of the focus of the focal reflector or refractor from the 

 points in which the ray touches the two caustic surfaces, are propor- 

 tional to the squares of the sines of the angles which the plane of 

 osculation makes with the tangent planes to the two developable 

 pencils. And when the ray is one of those principal rays, assigned 

 in the fourteenth number, (the focus of the focal surface being at the 

 principal focus corresponding,) then the contact of the second order 



