108 



-P = ?, + 2 



L [n+ 1]» 1 M^/+^)"+' [«+ l]H-i(z, + B)« + i 



„„ (« + "+^^^7^Vrf^A5^>' U,/ / . (B') 



+ ^(»^') o.'o [» + Ij-' + i [n' -t- IJ-Hri (2^ ^ ^)«+i (z, ^ ^)«'H 1 ' 



SO that we are conducted by this other method to the same expres- 

 sion (0) for the characteristic function of an ordinary optical sys- 

 tem, as that which we before obtained by performing the integra- 

 tions (N). In all these expressions the sign 2(„,„,o,'o° denotes a sum- 

 mation with reference to the variable integers n, n, from zero to 

 infinity. 



Case of a Plane System. 



5. A similar analysis may be applied to integrate the partial dif- 

 ferential equation 



to which the equation QA') of this Supplement reduces itself, when 

 we consider a system of rays of ordinary light, contained in the plane 

 o( xz. In this case, if we put 



X, = (.T — x,.) COS. XX, + (a — z,,) COS. zx, 1 



( (D') 



. Z^Z= (X — Xii) COS. XZ, -jr (Z 3„) COS. zz, -' 



t 



we may suppose x:, 2, to be new rectangular coordinates, in the same 

 plane as nz, and such that the axis of z^ coincides with the direction 

 of some given ray of the system : and we may denote by a, y, the 

 cosines of the angles which any near ray makes with these new axes, 

 so that 



We shall then have for one form of the integral of the partial diffe- 



