129 



(y + y){<^i'- ^^ + d^g-dx) + 2(rfy + dy'){dp.dx -f c?y.rf^) 

 + tf (« + »'). dx + rf'{,3 + /3'). rfy+ «r=(y + y'). rfa =0; 

 the condition (H") may therefore be thus written, 



d'*". dx + d^H". dy + d^-/. dz = d'^u.dx + <^/3. afy + d^y.dz : ( I'") 



besides, when the given ray, or axis of the system, is made the axis of (2), and when we take 

 for the two independent variables the two quantities (a, b), that is, the coordinates of the pro- 

 jection of the point in which the ray crosses the perpendicular surface, [57]. we have, from 

 [59.]. and from the nature of the functions «e", jS", y", 



d»»" = 0, d^^" = 0, rf V = «?*y. da — dx, dbssdy, 

 (PV dPV (PV 



^'"=1!^ • '^^ +2- rf-^i^ • ^"'^^ + dldf^ • '^^^ 



d^V (PV d^V 



dx^dy dx.dy^ ^ dy^ ^ 



so that (I'") becomes 



§- ''''•^''- ^, • '^'-'^'^'•^^- ''■'^' + w- '^ = '' ^^"> 



tliis then is the cubic equation which determines on the given mirror, the directions oi focal 

 inflection, that is, the directions in which it is cut by the osculating focal mirror ; and comparing 

 this with the cubic equation (F"') which determines the directions of spheric inflexion on the 

 perpendicular surfaces, we see that the planes which pass through these directions of spheric 

 inflexion, and through the axes of the system, pass also through the directions of focal inflexion 

 on the mirror ; so that the number of the latter directions is the same as the number of the 

 former. If then there be but one direction of focal inflexion on the mirror, that is, if the cubic 

 equation ( K'") have two of its roots imaginary, the principal focus is outside the little ellipses of 

 aberration, and the caustic surfaces cross the plane of aberration, in those two limiting lines, or 

 tangents to the little ellipses, which we have considered in [62.] ; but if there be three direc- 

 tions of focal inflexion, that is, if the three roots of (K'") be real, then the limiting lines of 

 aberration become imaginary, and the principal focus is inside the little ellipses. And if the 

 equation (K"') be identically satisfied, that is, if the mirror have contact of the third order with 

 its osculating focal surface, then the little ellipses themselves disappear, and the aberrations of 

 the second order vanish. 



