138 





A, = A(M, + __.„'+__. J. + 



aC) being, as in the preceding paragraph, the density at the point a =0, 5=0, that is, at 

 the point where the given ray meets the mirror: and substituting in these developments (Y'"), 

 the values of a, b, a',b', given by the series (T'"), we find that the two densities Aj, At, 

 are the two values of the following expression : 



A = A<^/") -j 2 — \r.{u-\-rM" + ... ± r^. (?<' + r.M"'+ )] 



■ ^ ^ '^ {zi=.r^- (to + r.w"-f ..,)+r.(w' -|-r,tu"'+.,.J] 



' db 

 + &c. 



which is of the form 



AS AV) + A&»'.r+ Aff).r*+... 



dbH.(AM+AJJ).r +...), (Z'") 



the coefficients being functions of the polar angle (v). 



Similarly, if we denote by y,, y^, the cosines of the angles which the two near rays, passing 

 through the two points {a, b), {a', b'), make with the axis of ^z), that is, with the given ray, 

 these cosines will have developments of the form 



z±zri. (rti). r+ rW. r' + ...) (A"") 



the coefficients being also functions of the polar angle (d) ; and the whole number of the near 

 reflected rays, which pass within the little rectangle (r.dr.dv) upon the plane of aberration, 

 being equal to the sum of the two values of the product y. A. SC), will be expressed by a 

 development of the form 



q;») = ri. dr.dv.(^Q(°> + Q(i). r+ Q(«). r^ -f- ... ) (B"") 



where QW = a"*', r/^**', and the other coefficients Q('', Qf'',... are other functions of the polar 

 angle (v), which may be determined by the formulae (X'"), (Z'"), (A""). Confining ourselves 

 to the first term of this development, and dividing by r.dr.dv, that is by the area of the little 

 polar rectangle upon the plane of aberration, we find the following approximate expression for 



