149 



ut, iKt, being rational and integer functions of z, not exceeding the <'* dimension ; so that we 

 may put 



Ut = Uifl + Ut,i . Z + «(,2 . «' + ... w<,<'. ^' + ••• tit J • «'. (A(*)) 



Wi=W,fi + lU/,1. 2 + W/,2. 2' + ... >»«,<'• z" + ... Wt^. z', 



"t,o> tk,i> ••• '"'1,0) 'wi.u ..• being constant quantities, not containing the polar angle v, and de- 



jjending only on the position of the given ray, and on the nature of the reflected system. la 



order therefore to complete our determination of the polar functions mW, wW, it is sufficient to 



calculate general expressions for the constants ut,f, wtjt', considered as functions of the indices 



dx' djf d^sf di/ 

 t, t', and of the partial differentials —r- , --r- , , „ , ..,—— ... ; since these differentials may, 



da do da^ da 



as we have before remarked, be deduced from the differentials of the characteristic function of 



the system. 



To calculate these constants, the method which first presents itself, is to substitute in the 



equations (1)">, (2)W, in place of «', u" , w', w", ... their values (Z""), (AC*)), and to compare 



the corresponding powers of z. Thus, if we confine ourselves to the constants ut^, tvi^t, which 



multiply the highest powers of z, as the most important in our present investigations, because 



when iu diminishes z increases without limit ; we are to retain only those values of jkC which give 



terms multiplied by z', and it is easy to see that these terms are distinguished by the relation 



2 + «j = 2£a + 2/3; (B(^) 



putting them, then, under the form fii^. to'+2. z', we have when t=l, 



fi-T- = -iTTM ' 2- ~-n:—-7r- («i.i = 



when t = 2, 



2. 



da^'d6^<^ ' da.db ' " da^'M^n ' ^''^ ~ da.db ' 



d^'+^^x' , d'x , d^x' , d'x* 



da^'db^^ da^ da.db ' '"' ^ '■^ db 



and, when < > 2, 



i^'^^^x' d^x' . , rfV 



i"M = -:rr7r • "'«-i.'-i +i--jiT- 2.w;,s.«'<_, (_, 



da^'db^^ '^' dadb —-'-•» dli 



die sums in the second members being taken from irrl, to«=:f — 1: and since the equa- 

 tions (1)W, (2) 'J, give, by comparison of the highest powers of z. 



