127 



(^y - Sx-) (cy - Dx') = (%' - Ciff, (E'") 



and which may easily be shewn to be the same with those limiting lines which we have already 

 considered. 



[64.] Secondly, respecting the surfaces that cut the rays perpendicularly, and which are 

 given by the differential equation 



ada + /3db + ydc = ; 



we have seen in a former section that the principal foci are the centres of spheres that have 

 contact of the second order with these perpendicular surfaces ; and if we wish to find the direc- 

 tions in which they are cut by those osculating spheres, we must express that the sum of the 

 terms of the third order in the development of the ordinate of the sphere, is equal to the corres- 

 ponding sum, in the development of the perpendicular surface. This condition, when the ray 

 is taken for the axis of («), gives, d^c = 0, that is, d^x. da -f- c?°/3. db = 0, which produces the 

 following cubic equation, (see 59.) v 



0=^. da^ + S.^^. da^M + 3.±^. da.m ^ -^. d5^. (F") 



This equation determines the directions of spheric inflexion upon the perpendicular surface, 

 that is the directions in which it is cut by its osculating sphere ; and the condition for there 

 being three such directions, that is for the three roots of this cubic equation being real, is 



^ dJ'V dW __ dH' d'V ■} 2 ^ / d'V Y d^V d^V ~i 



\dx^ ' dif dx^.dy' dx.dy'^ ^ ' Wdx'^.dy) dx^ ' dxdy'j 



<^ f d'F Y d'V d^V -y 



l[d^')~d^^- ~dp i ^°' ^ "^ 



that is, F" < 0. When, therefore, the principal focus is inside the little ellipses of aberration, 

 there are three directions of spheric inflexion on the surfaces that cut the rays perpendicularly ; 

 and when it is outside those little ellipses, there is but one such direction. It appears also, 

 from the formula (P"), that the aberrations of the second order do not vanish, unless the surfaces 

 that cut the rays perpendicularly have contact of the third order with the osculating spheres that 

 have their centre at the principal focus ; this condition is expressed by four equations which are not 

 in general satisfied : and for this reason I shall dispense with considering the aberrations of the 

 third order, because they only present themselves in some particular cases ; for example, in 

 spheric mirrors, the theory of which has perhaps been sufficiently studied by others. 



ZSS.'} I shall conclude this section by shewing that the conditions for contact of the third 

 order between the perpendicular surface and its osculating sphere, which, as we have just seen, 

 are the conditions for the aberrations of the second order vanishing, are also the conditions for 

 contact of the third order, between the mirror and the osculating focal surface, ( Section VIII.) ; 



VOL. XV. . T 



