-,57 



axis of z, and the principal focus for origin. Then, if we neglect the 

 squares and products of A«, A/S, vre find by the preceding theory, 



§ being the distance from the principal focus to the plane of aberra- 

 tion ; if, therefore, we suppose this distance f to be unity, and repre- 

 sent by a, b, the corresponding values of Aa, A/3, we shall have, 



A«= rf, A/3 = 6; (P*) 



and if we take the principal focus for origin, the coordinates of the 

 point in which the near ray intersects the plane of aberration will be 

 a, by 1. If now we conceive another plane of aberration, perpendi- 

 cular to the principal ray and passing through the principal focus, 

 we shall have, for this new plane, f = 0, and the expressions (0*) for 

 the components of aberration vanish : in this case, therefore, it is 

 necessary to carry the approximation farther, and take account of 

 terms of the second dimension, in the variations of a, /3, y. For this 

 purpose we may differentiate twice successively the equations (K), 

 as if a, /3, y, were independent, making after the differentiations, 

 X, y, z, ix', hy, $z, ^^z, each = 0, and changing 5a, 5/3, iy, 5'a;, 5'y, 

 to Aa, A/3, Ay, 2 Aa?, 2 Ay. In this manner we find' ' -'* > 



in which we may put 



(Q') 



(E*>. 



VOL. xvr. 



