84 Proceedings of the Royal Irish Academy. 



first approximation can be used in the terms of the third order. Hence 

 Y 



Z 



1 



Pi - /'i-l \ /' 



wliere 



- • Hi f*i-i) 



- \tu {,,tf + n,- 2 ) - f i i . 1 (»i in 2 + Ji.-.j 2 )} 

 = }/i,.i,., (/i,- - /uj., ' [(mi - m^)* + («i - />,-.,) 2 } 

 + it, (a*. - /u,-_, ' !(/(,»/, - /<,.,.''.■ i) 2 + Ov'i - /*.-i".-i) 2 }- 



The last line vanishes with /»,- For spherical surfaces. The errors in the point 

 of incidence Y.Zi are now of the fifth order, and will only affect the 



ulation of W{ and the optical errors of the ninth order. 



These indications will suffice to Bhow that the calculation of the nine 

 errors "i the fifth order for a symmetrical 0] i i--.il m -inn. though complicated, 

 would presenl no serious difficulty. Thai the calculation has not apparently 



n made doubt! tits from the practical difficulty of reconciling the 



removal of the errors of the third order with the conditions of achromatism 

 — a difficulty which hitherto has nol been completely overcome. 



1l'. It may be convenient to add a brief summarj of the contents of this 

 paper : — 



$ 1. The modified characteristic function. 



§2. I 'ti .it one Burface. Form of the second order terms 'I ,". 



§3. Introduction of new variables. Transformation of // ,". 



§ 4. Elimination i l-order terms, Calculation of .'/ . 



| 5. < lhanges in the coordinate 



6. E lion to any nu refractions. 



j 7. Formulae adapted t" ■<■ mirror system. 



Astigmatism and curvature of the field. The Petzval condition, 

 ry in most general terms. 

 § Hi. Sphei ration and coma. The sine condition. 



? 11. Further development. Preparation of formuhc For calculating 

 rs of the fifth order. 



