145 



3d, to effect corresponding transformations on the integral formulae (E"") (F"") of the preced- 

 ing paragraph; and 4th, to perform the double integrations within the limits of the question. 



In this manner we shall obtain developments proceeding according to the ascending powers 

 of the little circular radius (r), to represent the optical quantities which we have denoted by 

 Of.'''), SW ; it will then remain to suppose (r) infinitely small, and the resulting expressions 

 Q(''''\ iS(<'''), which must evidently satisfy the relation 



A "*> being the density at the mirror, will each serve to measure the density at the caustic sur- 

 face in the sense that we have already explained. 



(I.) First then, with respect to the coefficients «W, tu(*), of the series (T'"), 



a = ur -\. i/r^ + ... «('). r 2 + ... 



6 = wr'-J- ttj'r + ... wW. r a -|- ... ; 



it is evident that if we differentiate these series with respect to Vr, we shall have, supposing 

 ii/r to vanish after the differentiations, 



da „ cPa „ d^a „„ . 

 dj^r di^r dt/r 



and in general 





[«]* expressing, according to the notation of Vandermonde, the factorial quantity 1.2.3 



(s_l).j!. If then we differentiate, with respect to t/r, the equations 



r. cos. V = x', r. sin. v = i/, 



considering xf, y', as functions of a and 6, and these as functions of is/r; the resulting 

 equations, 



d'.r, cos. x> __ d'.af d'.r. sin. o _ rf».y AT""^ 



will serve, with the help of the formulae (S"")> *<> determine successively the coefficients u', w', 

 m", w", ...mW, tt)W, as functions of those which precede them; observing that the partial differ- 

 entials-^, -TJ-, ——,,..-4-, -^ , -^, ... are the coefficients of the series (S"') of 

 da do da^ da do da" 



x2 



