Aberrations of a Symmetrical Optical Instrument. 279 



The left-band side of equation (21) is — times the curvature 



fid 



of the image after the nh refraction. Hence for a flat 

 image we must have 



i n \fii {ii_ij 



the well-known Petzval condition. 



It remains to consider the defect of distortion due to 

 unequal magnification in the inner and outer parts of the 

 field. It is assumed that the image has been corrected for 

 central spherical aberration, coma, astigmatism, and curva- 

 ture, so that it now lies in the focal plane. We proceed to 

 obtain the condition that it shall be a faithful copy to scale 

 of the object. 



The refracted raj at the i'th surface is 



X-fi _ Y-,, _ Z -2> + ^) 



CI i 



This cuts the image plane Z = s/ where X=±a?/-f A#», 

 Y=#' + A#? (say). 

 Then we get 



and similarly for the z/'s. 



Rejecting terms of the fourth order this gives 



A*' «*'_ AV /, g,\ 



By (5') we have 



**/ _ JW As'/ 6y_ ^ A*/ J^ A t £ , 



inks; ( 1 fcfiK M „ 

 A similar equation follows for the incident rav. 



