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



Then multiplying equation (9) through by hi 2 , and since 



A s J = As. 



i+V 



2 v r J XfjLsJ 2 V r» \/a. ^_i/ 



By summation, since A5 1 = 0, 



? 2 



-K^)|«^©-(^)|Q.*.-A(i)(i + «,'.«) 



-!(X'+^)Sa(i)il + Q.;,rt)- + i(l'+,')2l(I--i). |16) 



When \ and //, are each zero the object and image are on 

 the axis. The aberration is then the central spherical 

 aberration only and is given by the first term of the right- 

 hand side of (16). 



The condition for no central aberration is thus 



S&W-^t-— )=0. . . . (17) 



Supposing this satisfied, let A, and fi be now small 

 quantities whose square may be neglected. 



Then the expression for the longitudinal aberration is 



\ III J l \fliSi fli-lSi/\ FlWp+1/ 



The evanescence of the coefficient of — — . ^ is the 



condition for the absence of coma, the balloon-shaped flare 

 produced in the image-plane owing to the images for 

 different values of ^ and 7j ± being distributed along the line 

 C t -K t -Kt + i in fig. 1. As in Whittaker's tract (' Theory of 

 Optical Instruments,' 1907, § 25) it may be shown that to 

 this order of approximation this is identical with Abbe's 

 sine condition. 



