Plummer — On the Symmetrical Optical Instrument. 81 



10. Thus the terms of the form SA'p t , q are independent of (iy, £/). They 

 disappear on differentiation, and have no effect on the image points. Similarly, 

 terms of the form SS'p/ 4 are independent of the position of the object 

 point. They constitute the whole of the spherical aberration. Terms of the 

 form 2£"/Oo%/ are linear in (>y , ZJ). On differentiation they give rise to a 

 pure distortion which affects the position, but does not disturb the quality of 

 the point image. 



The remaining terms, which are represented among the third-order errors, 

 and which have not been specially considered, are of the form ^F'pj''r (U : 

 When Z = they give 



Zj = - , h -2F'pj"> . Zqruy/pf. 



These terms represent coma. The corresponding image formed by any 

 zone of the instrument is the twice-traced circle 



[V -%{t- ZFp/« (1 + q)}f + Zf = (IqFp 1 '«)V- 



In the third order q = 1, and this circle is 



i n j- % (l-2F P ;))z + Zf = FWpJ 2 > 



and the whole system touches two straight lines meeting at 60° in the 

 first-order image. This is the characteristic error of the single parabolic 

 mirror for an object at infinity, and might be called the parabolic, as 

 distinguished from the spherical, aberration. 



If there is no spherical aberration, S contains no terms independent of 

 (t) , £ ), and therefore 



dS_ dS 



when ) /n = ^ = 0, ty = Zj = 0. These points on the axis are stigmatic points. 

 If there is no coma, S contains no terms linear in (t) , £ ). Then 



d 2 s d*s ' d 2 s d 2 s 



dv/dtio dii/dZo dZ/dtio 3&'3?i 



and therefore the point (>; + 8%, Z + S£ ) to the first order in 8i| , S£ 

 is represented stigmatically if i; = £ = is so represented. In this case 

 the points j; = Z = 0, iy = Zj = are aplanatic. The second condition 

 alone can be written 



9 V 1 = g £o' = 9 V = ^ -1 = (12) 



dr\j dry' dZ/ dZj' . 



