508 Mr. T. Chaundy: Method of Line- Coordinates for 



Again, f = —a'/m / = —a/m', since a is an invariant. 

 But al = nu — vm = nu, since 7n = 0. 



Thus 



%'=—nu/lm' 



~ I ' Q + ^{Bn 2 -2Dnv + F(u 2 -^v 2 )} 



= -njlQ + n{Bn 2 -2Dnv + F(u 2 + v 2 )}/2Q 2 . 



In order that all the emergent rays may converge to a 

 common focus we require that £', f be each independent of 

 u, v. 



This necessitates 



DS-JQ=0, FS-LQ=0, D = 0, F = 0, 

 i.e. D = = J = F = L. 



In virtue of the identity DS + FR- JQ-LP=0 already 

 stated (18), it is clear that these conditions, apparently four, 

 really represent three independent conditions alone, i. e. two 

 in addition to the condition of no spherical aberration. 



When these conditions are not satisfied, we can state the 

 aberrations under the guise usually recognized, namely, error 

 against the sine-condition and astigmatic error, in terms o£ 

 our aberration coefficients. 



With light parallel to the axis, the sine-condition requires 

 that the ordinate of the incident beam bear a constant ratio 

 to the sine of the inclination to the axis of the emergent 

 beam, i. e., in our notation that vjn' be the same for every 

 ray. We may measure error against the sine-condition by 

 variation in v\n' between marginal and axial rays. 



Now for an incident paraxial ray, i. e. for m = = w, 

 we have 



n' = Qv + it(u 2 + v 2 )v, 



so that v / n > = i/Q _ Y(u 2 + v 2 )I2Q 2 . 



Thus error against the sine - condition is measured by 

 -A 2 F/2Q 2 . 



Satisfaction of the sine-condition requires F = 0. 



The third component of departure from point-to-point 

 accuracy usually considered is astigmatism. This is an odd 

 sort of quantity, having little organic relation with the 

 system, since it is necessary to specify the position of a stop ; 

 its discussion is correspondingly awkward, and I accordingly 



