163 



rection parallel to the axis of y", and then projected on the normal ; it is sufficient to change 

 the fractional powers of y". sin. v' to those of S' -f- y"^. sin. d', in order to obtain developments 

 for a and b, which shall satisfy the condition of the question, containing no negative powers of 

 any variable quantity, but only positive and integer powers and products of x" and of ■•£'. 



To demonstrate this theorem, let us resume the equations (GW), putting them by (U®) 

 (AW) under the form 



tP-i tan. ti'. x" = a — F{a, h) ; s. sin. v'. y" = b^ — 2<I>(a, b). (DW) 



Conceive a parallel to the axis of y", drawn through the point x", y", upon the plane of aber- 

 ration ; this parallel will meet the section of the caustic surface in a point having for coordinates 

 x", y"„ and the ray which has that point for focus will cross the perpendicular plane at the 

 mirror in another point whose coordinates may be called «„, b^ ; to determine these coordinates 

 we have by (DP)), 



• 8 P-i tan. ■!)'.«" = ao — F(„ « sin. v'. y'o = 6% — 2*0, {E<V) 



Fo> %> representing for abridgment the functions F{ao, bg), *((!„, b^) ; we have also, by the 

 nature of _y"o, 



da^ dbo db^ db^ V da J V " dbj db^ da^ ' ^ ' 



from which it follows that the locus of the point Oq, io, on the perpendicular plane at the mirror, 

 has for tangent the right line 



and that we can develope a^, b^ in series of the form 

 ao= iP~^ tan. »'. x" + 



This being laid down, let us subtract (ECJ) from (DC)) ; we find 



^ 



(HW) 





(1(7)) 



'^"» '^*o ■■ ■ rfflo^^^o'"' ['"}'" [»»']" 



and therefore, by (F^), 



VOL. XV. 



