40 



"idly d«dy ' J^Jy "" S/33y ' Sy» Sy* ' 



of which the three latter result from the three former. These six' 

 equations, which are all expressed by the one formula (t/') or (Z'), 

 provided that we consider 3a, 3.5, ^y, as independent, will give in 

 general a finite number of real or imaginary values for a, /3, y, fi, 

 and thus will determine a finite number of isolated points, as the in- 

 tersection of the caustic surfaces. We shall call these points the 

 Principal Foci ; and the rays to which they belong, we shall call the 

 Frincipal Rays of the system. In general, whether i/' be greater or 

 less than zero, we may employ the equations (T"') to determine a 

 finite number of isolated points and rays, to which we shall give the 

 same denominations. It results from the equations by which these 

 points and rays are determined, that if the focus of an osculating 

 s^'stem be placed at a principal focus of a given system, the oscula- 

 tion of the second order will be most complete, since it will be inde- 

 pendent of the direction of the plane of osculation (B") ; the three 

 first terms of the two developments in the eleventh number, namely, 



fV' + iW + irfV', 



becoming equal, independently of the ratios of 3«, 3/3, 3y. The prin- 

 cipal foci of an optical system possess many other remarkable proper- 

 ties, some of which we shall indicate in the course of the present sup- 

 plement. 



