126 



2 being the area of the ellipse, and (^|') an angle whose cosine, multiplied by the focal distance 

 of the centre of that ellipse, is equal to the semidiameter whose prolongation passes through 

 the focus ; we have therefore 



and the entire space over which the intermediate rays are diffused is 



2 + 2'=i[v'F + {^ — ^)[C(C — A)+B(B-D)]j6\ (B"') 



[63.] We have just seen, that in investigating the aberrations from a principal focus, it is 

 necessary to distinguish two cases, essentially different from one another. In the one case, all 

 the rays that make with the given ray angles not exceeding some given small angle {6), are dif- 

 fused over the area of a little ellipse; in the other case they are diffused over a mixtilf- 

 near space, bounded partly by an elliptic arc, and partly by two right lines, which 

 touch that elliptic arc, and which pass through the principal focus. The analytic distinc- 

 tion between these two cases depends on the sign of a certain quantity F", which is negative 

 in the first case, and positive in the second. It is therefore interesting to examine, for any 

 proposed system, whether this quantity be positive or negtive. I am going to shew that this 

 depends on the reality of the roots of a certain cubic equation, which determines the directions 

 of spheric inflexion on the surfaces that cut the rays perpendicularly ; I shall shew also that the 

 sign of the same quantity, is the criterion of the reality of the roots of a certain quadratic equa- 

 tion, which determines the directions in which the plane of aberration is cut by the two caustic 

 surfaces. 



First then, with respect to the caustic surfaces, it may be proved, by reasonings similar to 

 those of [61.], that the two foci of a near ray have for coordinates 



x', y, being the coordinates of the point in which the near ray crosses the plane of aberration, 

 determined by the formulae (V"), and ({,) having a double value determined by the following 

 quadratic equation 



(Is/ + ^«/ +M) (h + c», + Da)- cs«/ + cny = 0, 



in which A, B, C, D, have the same meanings as in the preceding paragraph. The intersection 

 therefore of the caustic surfaces with the plane of aberration, is to be found by putting j,.= Ot 

 which gives :* = {, x,-=: x', y, = y', 



(Ace, + Bfi,) (Ccc, + Dl3,) - (5=c, + C/i,Y =0; (C'l 



the condition for the roots being real, in this quadratic (C"), is 



(AD—BC)'—i{B'' — AC)(C''—BD)>0, (D^ 



that is, F" > 0, so that unless this condition be satisfied, the caustic surfaces do not intersect 

 the plane of aberration ; and when this condition is satisfied, the intersection consists of two 

 right lines, which are determined by the equation 



