(GM) 



158 



,. . c?'* d'<S> ePF d'F , , ^ „ . , . 

 and if we eliminate — ; — , , „ , —rv > -; — rr . by the following relations 

 rfa' da.do da^ da. do 



^ ~ ~db^' ' da"'.db"" "*" da"'.db'^ f 



_d£_ rf'W+w'j. _ ^^ ^'>» + "''^. tf^'+ '^V ^ 



c?a ■ da'".d^ "5F " da"Kdb"'' da"'.db"'' ^ 



in which m, »»', are any positive integers satisfying the condition (L(f ), we arrive again at the same 

 expressions for the constants of the form m^,(, Wtjt, as those given by the equations (l)i,i, (2)i,i, 

 (1)2,2) (2)2^, (C(''), which we obtained before by reasonings of so different a nature. It results 

 from these equations, that if in the developments (Z(^)), for the components of aberration at 

 the mirror, we confine ourselves to the terms of the form 



«2,+l, 2r+l • «^'+^ «^-'^- ^r^'^^ . t.2.,2. uKu>'-K ^r^'+S 



which correspond to the greatest negative powers of w, or of the sine of the polar angle v ; the 

 sums of these terms, taken fi"om t = 0, to t = 00 , may be calculated by the binomial theorem, 

 and are thus expressed : 



We shall return to this remarkable result, and examine its optical meaning. 



As another application of our general formulae for the constants m, („ tUn, , let us take the 



terms of the form 



which correspond to the next greatest negative powers of w, or of the sine of the polar angle v, 

 in the development (Zt^)) for b, that is, for the aberration at the mirror measured in a direction 

 perpendicular to the tangent plane of the caustic surface, and considered as depending on the 

 polar coordinates r and v, which determine the magnitude and direction of the aberration on 

 the perpendicular plane at the focus. We have, by what precedes, 



«'2r+.2, 2.+I =2Hr^2, 2,^1 +2.«.<';''^2,2r+l 



= [or^2'+»...[or'.'no]-(2'"-2'+ 1) [«'_r+4]»'__^,,^,^^; (k.^ 



and the summation here indicated, with reference to the variable integer n', may be performed 



