78 Proceedings of the Royal Irish Academy. 



8. The theory of the errors of the third order is contained in the 

 equations 



a w* _ ? w ic 



Wo- ?i(/ . &-?.= a ^r. 



- 4 IF" = -W + B,,/* + ICtJ + 2D PoP / - 4^p t r ti - ±F Pj \, 



and their effects in the image plane follow immediately. But it is convenient 

 to consider the incidence of the ray on a parallel plane at a small distance e 

 before the image plane. The displacement of the point of incidence is given 



by 



-\ l;' ~ k/'JwXLj-it, 



*" ■ „ ,-,*(£,- A'y) //, -^(Lf-Kf)- 



The second term depends only on the object-point. It represents a linear 

 distortion, or, in other words, a Bimple change of scale value. Therefore it 

 needs no farther consideration. Let the ray now meet a sphere of radius ( >, 

 such that the image plane is the tangent plane at the point where it meets 

 the axis. Then 2/w yf when -., - 0,and the first term becomes 



■:,,-. \L K,)H,p " 2w?(Zj-Kj)p 



"iP- 



The assumption that :., - 0, and therefore :. =0 to the first order, does not 

 cl the generality. Similarly 



AZi = - T)„%'/2, tj p. 



Now the errors which have the same form are those multiplied by 

 C and /'. When Z,, - they are 



m n (20 h /> >,'.,/ 



:, D „ ; . 



If p = p l where 



1/p, = 2 W (2C+ D) 



the image clearly becomes a straight line transversal to the axis, for 



i/ '/ A >i, = 0. 



