140 



JS r.drdv, which represents the assigned space upon the plane of aberration, and the extreme 

 values of (u) being supposed such as not to render the series (T'") illusory. These series (T'") 

 serve also to correct the approximate expression of the preceding paragraph, for the first term of 

 (a) ; which first term was there taken as 



0=:--^, (Q'") (67). 



whereas by employing the remaining terms in the developments of x' and y, we have now found 

 it to be 



(Be , . sin u — Q , . cos. v \ 

 - — ci ^= 



a 



a value which differs from the preceding, by the addition of ( 5,'- )• And if, by means of 



this corrected value, and by using as many of the remaining terms of (T'"), as the question may 

 render necessary, we eliminate (r) and (u) from the polar equation of any given curve upon the 

 plane of aberration ; for example, from that of the boundary of the space// r.«?r.rfu, for which we 

 have already determined the corresponding quantity of^light, and the area over which that quan- 

 tity is diffused on the perpendicular plane at the mirror ; we shall find the approximate equa- 

 tion of the boundary of this latter area, and thus resolve a new and extensive class of ques- 

 tions respecting thin pencils, for which the formulae of Section IX. and those of the 60th para- 

 graph would be either inadequate or inconvenient. 



As an example of the application of the reasonings of the present paragraph, let us conceive 

 a small circular sector, upon the plane of aberration, having its centre at the focus of the given 

 ray, and having its radius {r) so small, that we may confine ourselves, in each development, 

 to the lowest powers of that radius. Let (4-) denote the semiangle of this sector, and let (v") be 

 the polar angle which the bisecting radius makes with the axis of(x'); then v" — -i^, ^''-f-i^, 

 will be the extreme values of the polar coordinate {v), while the corresponding limits of the ra- 

 dius vector will be (o) and (r). Denoting by (SW) the whole space occupied on the perpendi- 

 cular plane at the mirror, by the rays which pass within the given little circular sector, and by 

 (Q'"^') the number of these near reflected rays; the formulae (E"") (F"") give, for these quantities 



S^-^) = //f7W, rl. dr. dv-^.A.f [/(<». dv, 



QW =ffQPl ri. dr. dv = f . rf . / Q(«'. dv. 

 or, substituting for [/("', Qt**) their values. 



SW=Hiil££f. f. '^" 



3.iV^C' '/. 



sm. t) 



S.J.VtC ' -/sin.i; 



(G"'') 



