172 



remarkable relation, that the segments into which the principal focus divides that diameter of 

 such an ellipse upon which it is situated, are proportional to the squares of the semiaxes of the 

 lines of uniform density ; in such a manner, that when the principal focus is situated at tlie 

 centre of the ellipses of aberration, the excentricity e vanishes, and the lines of uniform density 

 become a series of concentric circles ; and when on the contrary, the principal focus is on the 

 circumference of the ellipses of aberration, then e becomes equal to unity, and the lines 

 of uniform density become a set of rectilinear parallels to the axis of y, which axis in this case 

 coincides with the common tangent to the little ellipses of aberration, drawn through the prin- 

 cipal focus. When this latter circumstance happens, the two expressions (Y'^') for the density 

 at this principal focus, coincide with one another, and become 



7 



-^ = 00 ; (Z(8)) 



in this case therefore, we should be obliged to have recourse to new calculations, and to 

 introduce the consideration of aberrations of the third order. We may remark that the quan- 

 tity F", the sign of which distinguishes between the two chief cases of aberration from a 

 principal focus, becomes = 0, in the case which we have just been considering ; and since, by 

 Section XII., the sign of this quantity JF" determines also the nature of the roots of the cubic 

 equation 



which by the same section assigns the directions of spheric inflexion upon the surfaces 



of constant action, and of focal inflexion on the osculating focal mirror ; it follows that 



in the present case this cubic equation has two of its roots equal, and therefore that 



two of the directions of focal or of spheric inflexion coincide. With respect to the value 



of these equal roots, we have from our present choice of the coordinate planes the equations 



d'V d^V 



A=0, B = 0, and therefore by [62], —-— = 0, , „ , = 0: thus the cubic equation be- 



comes 



d^V d?V 



from which it follows that the two directions of inflexion which coincide with one another are 

 contained in the plane of xz, that is, in a plane passing through the axis of the reflected system, 

 and cutting perpendicularly the lines of uniform density. 



[76.] Many other remarks remain to be made, in order to illustrate and complete the theory 

 of the present section ; but as we shall have occasion, in treating of refracted systems, to resume 

 this theory under a more general point of view, we shall only here add, that the function 

 which we have called the density, may differ sensibly in many instances from the observed in- 



