56 



the plane of aberration. Let .t, y, z, be the coordinates of the 

 given point, and x + Ax, y + A?/, z + A2, the coordinates of the 

 point in which a near ray is cut by the plane of aberration, A being 

 here the mark of a finite difference ; we shall have the condition 



= aAx + /3Ay -f yAS, (L*) 



a iS 7 being the cosines of the angles which the given ray makes with 

 the axes of x, y, z : and if we determine the successive differentials 

 of x, y, z, with reference to a, /3, y, by differentiating the equations 

 (C) or (X) as if a, /S, y, were three independent variables, and by 

 putting 



= oe3x + 0,'iy + yJz, 



O^uVx + fiPy^-yVz, . ^^^^ 



&c. 



we shall have 



ACT = [Jx] + i [3'x] + -^ [S'x] + &c. 



Ay = [Sy] + 4 [J'y] + -^ i'^V^ + &^- > (N^) 



Az = [Jr] + f [3«z] + -ij- [J'z] + &c. 



the expressions \}x\ [5*a;], &c., being formed from ix, d*x, &c., by 

 changing the differentials ^a, 5/3, 3y, to the finite differences Aa, A/3, 

 Ay : and finally, the lateral aberration of the near ray will have for 

 expresssion 



4/ (AX)* + (Ay)" + (Ae)' 



Let us apply this general theory to the case when the ray from 

 which the aberrations are measured, is a principal ray of the system : 

 and in order to simpUfy the calculations, let us take this ray for the 



