408 Mr. T. Smith on tie 



that there should be no term in r on the right side of 

 equation (4). 



The substitution of the above values for a and ft in (3) 

 yields a value for the departure from the sine condition 

 which should be comparable with Mr. Allen's second con- 

 dition. The fact that their character is essentially distinct 

 shows that one of them at any rate is not the sine error. As 

 a matter of fact his second expression is comparable with the 

 coefficient of S in (2). One of the factors multiplying (2) in 

 the complete expression for the comatic displacement of a 

 ray is 1 — 5, and the coefficient of S therefore takes the place 

 of (2) as the important factor in the value of the coma when 

 the aperture stop is in contact with the objective. The 

 statement that this expression only measures i( the amount 

 of coma provided there is no spherical aberration " is 

 incorrect. 



The expressions quoted above for spherical aberration and 

 coma hold for any thin system, no matter how complex its 

 structure may be, and whether the surfaces are cemented 

 together or there are air-gaps. It is a simple matter, if 

 desired, to introduce additional variables to show the effect 

 of varying the curvature differences bounding these gaps. 

 There will be no change in -or, but a and ft will be respectively 

 quadratic and linear functions of such curvature differences. 

 This follows at once by noting that such gaps are created by 

 bending part of the system relatively to the rest, thus causing 

 alterations in the aberrational coefficients of the two parts of 

 the kind indicated in equations (4) and (5). The additional 

 coefficients in a and ft are necessarily of a symmetrical form. 

 When the objective is a doublet with one air-gap, one * 

 additional coefficient will occur in the general expression 

 for a, one in the expression for ft, and one in the formulae 

 for the curvatures of the system in its zero conformation. 

 Thus in the most general case considered by Mr. Allen 

 only seven quantities are needed in place of the fourteen he 

 tabulates. As a rule, however, there is not much point in 

 taking the air-gap into consideration as a separate variable. 

 It is usually possible to employ cemented objectives, and in 

 most instruments this is very desirable on account of the 

 better light transmission so obtained. 



The reduction in the labour of calculation obtained by the 

 arrangement described above does not exhaust the advantages 

 of the system. If a triple lens is to be calculated in place of 

 a doublet the a and ft of the triple objective may be derived 



* If g is the gap the coefficient of g 2 in a, is one quarter the coefficient 

 ofr 2 . 



