406 Mr. T. Smith on the 



Formulae for the purpose of such calculations, though not 

 usually expressed in a form resembling that adopted by 

 Mr. Allen, have been known for many years. Although a 

 simple calculation by known formulae would furnish equi- 

 valent information, it was considered desirable, in view 

 of the special circumstances existing at the time, to 

 publish tables in the form of those issued by the National 

 Physical Laboratory. These appear to have served satis- 

 factorily the limited function for which they were intended, 

 and any value they still possess may be regarded as 

 accidental. 



It has been customary for a maker of telescopes or other 

 optical instruments, when working out a new objective, to 

 rely upon his previous experience to enable him to set 

 down approximate curves on which to base his calculations. 

 Under favourable conditions a very limited amount of trigo- 

 nometrical ray tracing enables him to reach a satisfactory 

 final solution. This method works satisfactorily in expe- 

 rienced hands, but such experience becomes quite unnecessary 

 if other methods are adopted. Without any experience 

 whatever it is possible, with the aid of a little algebra, to 

 obtain in a few minutes an approximately correct form for 

 an objective provided the conditions to be satisfied are stated 

 in a suitable form. Mr. Allen's formulae enable such cal- 

 culations to be made, but they are cast in a form which 

 involves an unnecessar}^ amount of arithmetical work. 

 Fourteen coefficients occur in his two expressions for the 

 spherical aberration and the sine error. It is obvious that 

 many of these do not involve separate computation — for 

 instance, several identical relations exist between A, B, C, 

 D, E, F, P, Q, and R. It seems preferable to express the 

 fundamental quantities in a form which takes advantage oE 

 these relations. 



The writer has pointed out elsewhere * that all the first 

 order aberrations of any thin objective for light of a given 

 wave-length are determined by three quantities which depend 

 upon the refractive indices of the glasses and the curvatures 

 of the surfaces, but not on the position of the object. If the 

 three quantities be denoted by a, /3, and -57 1 the factors 

 which involve the constructional data of the objective in the 

 expressions for the spherical aberration, the coma, and the 



* Proc. Phys. Soc. vol. xxvii. p. 485. 



f In the standard notation these quantities are denoted by 4C + 2t»r + l, 

 B'— B, and w respectively. 



