278 Mr. E. Howard Smart on the Third-Order 



So that the separation of the primary focus from the Gauss- 

 image plane after refraction at the ith. surface is given by 



^=-!(f)>'<«) 



+i\*i -(-- -L\ . . (i9) 



The separation of the secondary focus is easily written down 

 from (16). Denoting this by s A«i' we have 



+ Jx»si-(i— iV (20) 



* i ri\fii Hi- 1 l 



Assuming, then, the errors of spherical aberration and coma 

 to have been eliminated, it is clear from equations (19) and 

 (20) that the primary and secondary foci can only be brought 

 into complete coincidence if also 



|a(I) ( i + qa*2)*=o. 



This term occurring as the coefficient of V, and therefore 

 depending on x 2 , indicates that the corresponding astigmatism, 

 which the evanescence of this expression removes, is due to 

 the outer parts of the field. 



It may be further noted, from equations (16), (19), and 

 (20), that when central spherical aberration, coma, and 

 astigmatism have been corrected the image still suffers a 

 displacement from the focal plane given by 



H-i 

 Hence 



ft 



s i+i 2 i Ti Vi Mi-^i/ 



^= : ii(I A-). . . . (21) 



