161 



of (L"") [68.] have for expressions 



8. sin. (« — v) _i_ // • \ 



u = — -—^ -. — , tti = ± ^(j. sin. v), 



P. COS. v' 



(U(6)) 



V being the polar angle, and e, P, v' constants which enter into the equations of the curves 

 (M""). Substituting these values in the series (T'") which may be thus written. 



(V(6)) 



and observing that as the negative powers of w are all odd, those of (sin. v) are all fractional, 

 we find the following transformed developments : 



^ or, / • vr+l '■+' T>_,'/sin.(u — v')\i' 



(.nO '' Vcos. vf. sm. v/ 



W" '^ 2r+l,<'-^^ Vcos. V' sin. «; 



A = 2(^3^ (.r sin. vY+\ 2,,;+\ ^^^^ ,. p-^'p'"' <"-^) V' 



\cos. «'. sin. D / 



=' V," <-''-)'*'-(0"-.„.- ^" (;£|iS) 



J 



(WW) 



which have the advantage of exhibiting to the eye, the manner wherein the rectangular com- 

 ponents a, b, of aberration at the mirror, depend on the polar components r, v, of aberration at 

 the caustic surface. To eliminate from these developments ( WC^)) the negative powers of (sin. v), 

 without introducing those of any other variable, or the positive powers of any quantity which 

 (like the z of preceding problem) becomes infinite when the polar radius r assumes a particular 

 direction ; let us resume the summations, expressed by the equations (HC')). It results from 

 those equations, or from the formulae (C(5)), (F(6)), on which they were founded, that if, in order 

 to begin with the greatest negative powers of (sin. v), we reject at first all but the greatest 

 values of t' in the developments (WW), namely i' = 2T + l in a, and t' =2r in 6, and denote 

 by a,, 6,, the sums of the terms that remain, we shall have 



a, = 



d'^F r. sin. (v — d') § 



da.db 



P. cos.!)' 



in which 



6 = r. sm. V + { —— -f / ] \ I 



^\da^^ {da.dbJ ) \ 



rf'*^^ /VW. sin. (d_i;')V 



P. cos. V' 





(X(6)) 



(YW) 



F, <!>, having the same meanings as in the foregoing problem. To find the optical meaning of 

 the binomial function (€), let us consider the points upon the plane of aberration for which that 



