170 



in which J5' — j4C, C^ — BD, will both be positive, and proportional to the squares of the 

 seraiaxes of the eUipses of uniform density. Multiplying this expression by r.dr.dv, and inte- 

 grating from r = to r ■=-r, and from c = to r = 2w, we find for the whole number of the 

 near reflected rays that pass within a small given distance {r) from the focus, the following ap- 

 proximate formula: 



ff A C) rdrdv = 2 A (/'^.rJX —^p^^ .^. . ^ "- ,„— ^ ^^— , , (P®) 



''•' ^ ^ g^V(i3^ — ^C).sm. i;-j-(C2 — BD) cos Jvl 



a transcendental of known form, which can be calculated either by elliptic arcs or by series. 

 And if we denote this transcendental by T, and reason as in [69], we find the following ex- 

 pression for the density at the focus itself, as compared with the density at another point of the 

 same kind, 



ir. Case. F" > 0, 



[74.] Let us now consider the case where F" > 0, that is, where the principal focus is out- 

 side the little ellipses [62}. In this case the points in which the near reflected rays intersect 

 the plane of aberration, are all comprised within the angle formed by the two limiting lines (Y") 

 [62], namely, the common tangents to those ellipses of aberration ; and if we tske the bisector 

 of this angle for the axis of x, the relation (N('^) will reappear, and the equation of the limiting 

 lines will become 



^^C^BD_ C D 

 x' AC — B' A B ^ ' 



Moreover, the rays that pass inside any little rectangle dxjii/, within the angle formed by these 

 limiting lines, are at the mirror diffused nearly perpendicularly over four little parallelograms, 



which are eqnal to one another, and have their sum = — ' ' , M being the same function 



as in [72] ; we have, therefore, for the density at the point xy, tloe following approximate 

 expression, 



aud the lines of un^m dendty are still given by the equation 



M = const, 

 which now represents a series of concentric aad similar hyperbolas, having the principal focus 

 for their common centre, and the limiting lines of aberration for their common asymptotes. 

 And if we multiply this expression for the density a/*' by rdrdv, and integrate from r = to 



