412 On the Correction of Telescopic Objectives. 



third term in the sine expansion, amount to the values given 

 at the head of the respective columns. 



Error. . . . -000005 -00001 -00005 -0001 



(a) 0°11' 0°15' 0°34' 0° 48' 



(b) 1°46' 2° 15' 3° 50' 4° 50' 



(c) 6° 0' 7° 8' 10° 40' 12° 41' 



(d) 10° 49' 14° 56' 17° 9' 23° 40' 



It is evident that the figures in the last row correspond to 

 rays which are very far from travelling within a capillary tube 

 lying along the axis of the lens. The figures of row (b), on 

 the other hand, indicate that the number of figures used for 

 trigonometrical calculations may frequently involve the 

 neglect of aberrations altogether, though these would not 

 be omitted in the corresponding algebraic operations. The 

 table shows that the statement made above in discussing the 

 reliability of results derived from algebraic calculation should 

 occasion no surprise. It is, however, important that more 

 should not be read into that statement than it contains. The 

 field over which such calculations are reliable does not extend 

 to the limits given in line (d) of the above table or indeed to 

 line (c). The neglect of the third term in the cosine series 

 (not the sine series) defines the theoretical limit of accuracy, 

 but this limit is not applicable to the algebraic expansion for 

 a series of surfaces owing to the neglect of product terms 

 which are not necessarily of little account. Mr. Allen has 

 simply taken the traditional view of algebraic calculations 

 without investigating its accuracy. The true position I 

 believe to be that the importance and reliability of algebraic 

 calculations in the determination of aberrations has been 

 underestimated, and that of trigonometrical work as it is 

 usually carried out overestimated. In both methods of cal- 

 culation it is desirable to employ about two more figures 

 than can be said to correspond in the final rays with the 

 mechanical accuracy attainable in the concrete instrument. 

 When the calculations are completed the last two figures 

 may be neglected. The reason for this is that aberrations 

 are eliminated by opposing aberrations of different signs 

 and necessarily large magnitude. A typical illustration is 

 afforded by the values found by Mr. Allen for the coefficients 

 L of a doublet and of one of its components. It is an 

 instructive exercise to carry out calculations for corrected 

 systems retaining in turn four, five, six, and even seven 

 figures. The values of the outstanding aberrations given by 

 the earlier calculations will occasionally be found to require 

 appreciable modification. 



