109 



rential equation (C), the following combination of equations: 



dU . dU _ 



between which a is conceived to be eliminated, and in which 



(F') 



' + ^' ^ 2.4.6... (2j + 2) 2i + 4 ' 



W{^), WiJ^%, being constant coefficients in the development of the 

 function W, according to the powers of a, and A being another con- 

 stant in that development, namely, the distance upon the given ray^ 

 from the point where it touches the caustic curve of the plane sys- 

 tem, to the origin of x^ and z^ . The first of the two equations C^O 

 becomes 



<;> 

 «« = «/ + 



when we put 



and gives therefore, by the well-known theorem of Lagrange, for 

 functions of a single variable, 



d" dF, /dtp,Y+l 

 ictj"' d»i \d», J 



Fix) =:F, + .^„, - ^f< f/,r'/,,„,, . (G') 



F(ot) denoting any function of a, which admits of being developed 

 according to positive integer powers of «, , and F^, <p^, being formed 



VOL. XVI. 



