136 



and the coefficients being successively determined by equations of the following form, 



dx' (!'■ t/ 



(1)W...0= ■^. «»-+--^. w.icC) +A»„ (V" 



(2)(').,.o = 4y|' • w-»«w + m^, , 



do 

 Uf^ ,.»•«, ^"g. r 2 , representing for abridgment the sums of the known terms of the di- 



mension 



[ -^ 1 in the expansions of j/ and y', according to the powers of r, obtained by 



substituting in (S'") the assumed developments (T'") in place of <z and i. The quantities 



iWj, AWj, are therefore rational functions of the preceding coefficients u, u', m('~'), xk, 



to', «)('-*', and therefore finally of « and tv; and these functions do, or do not, change sign 



along with w, according as t is an odd or an even number. Hence it follows, that the developments 

 (T"'), which represent the coordinates of the points, where the near rays passing through any 

 assigned point upon the plane of aberration are intersected by the perpendicular plane at the 

 mirror, are of the form 



a = r.{u + r.u" + rK «""+..,)+ ri. {u' + r.u"' + ) 



b = ± rh(u) + r. w"+ r\ w"" + ...)+r.(i«'+r. t«"'+ ...), 

 the coefficients ofthe fractional powers being real or imaginary, according as w is real or 

 imaginary, that is, by (U'"), according as [ ' } is positive or negative; or finally, by [61], 



according as the assigned point (r, v) upon the plane of aberration, is, or is not, situated at that 

 side of the tangent plane of the caustic surface, towards which is turned the convexity of the 

 caustic curve. However, when the polar angle (v) approaches to (0) or (180°), that is when the 

 right line joining the focus of the given ray to the assigned point upon the plane of aberration, 

 tends to become a tangent to the caustic surface, the numeric value of (sin. v), and there- 

 fore of (h)), diminishes indefinitely ; and consequently the coefficients which contain negative 

 powers of that quantity, increase without limit, so that the series (T'") become at length illu- 

 sory. In this case, therefore, it becomes necessary to have recourse to new developments, 

 which will be indicated in the succeeding paragraph. But abstracting for the present from this 

 case, which in examining the variation of the density of the reflected light upon the plane of 

 aberration, may usually be avoided by a proper choice of the focus from which the aberrations 

 are to be measured : it may easily be shewn, by reasonings similar to those of the preceding 

 paragraph, that if we consider any infinitely small polar rectangle upon the plane of aberration, 

 having its base = r.dv, and its altitude =: dr ; the rays which pass inside this little rectangle. 



