110 



(da\ / db \ da db ^ 



which determines the two foci of a given ray, and in which the coefficients , — , , 



da, d^ dec 



— , are connected by the following relation, deduced from the same section, 



-'■(^-^)-<-+v>4:+<^-+v')4=<'- <■") 



The condition of equal roots in (Q), is 



/da dbV da db _ 



this then is the equation which determines the relation between a, /3, that belongs to the 

 rays passing through the intersection of the caustic surfaces ; and it is easy to prove, by 

 means of the formula (R'), that it resolves itself into the three following, which however, in 

 consequence of the same formula, are equivalent to but two distinct equations : 



da db da db 



d/i da ' da dfi ^ ' 



The rays determined by these equations, I shall call the axes of the reflected system, and their 

 foci, for which 



da db _. 



I shall call the principal foci. 



[47.] We have seen, that a given ray has, in general, an infinite number of virtual foci, cor- 

 responding to the undevelopable pencils, and determined by the formula (P'), 



2'= J , . COS. ^X -f" €»• 8'"* '-^> 



and an infinite number of osculating foci, corresponding to the osculating focal mirrors, and de- 

 termined by the formula (C) 



— = — . COS. ^■^ ^ . sin. '4-. 



But when {g = ?i, that is, when the ray is an axis of the system, then the variable angles dis- 

 appear from these formulae, and all the virtual and all the osculating foci close up into one single 

 point, namely, the principal focus corresponding to that axis. Hence, and from the coefficient 

 of undevelopability vanishing, it follows, that each axis of the system is intersected, at its own 

 focus, by all the rays infinitely near ; and that this focus, is the focus of a focal mirror, which has. 



