438 



SCIENCE 



[N. S. Vol. L. No. 1297 



main only two unknowns to be determined 

 by conditions which will eliminate aberra- 

 tions. For the first, onr authors select color 

 — chromatic aberration. The second condi- 

 tion in one case that for spherical aberration; 

 in another, for coma; and in a third case, 

 for equality of three radii instead of merely 

 two. Evidently therefore this publication, 

 though valuable as a first, is only the first 

 among a large number of desirable thesauri 

 for optical designers. 



Of two solutions for the same physical 

 condition, equally correct mathematically, 

 one may prove in practise far superior. 

 Tables A and B enable us to compare these 

 two, both free from spherical aberration, 

 thirty-six samples of each. To the cautious 

 tyro, and also, it appears, to the expert, it 

 seems better to select surfaces of small curva- 

 ture where possible; although in microscopes, 

 as Abbe demonstrated, such counsel is often 

 misleading. Taking as unit the focal length 

 of the combined lenses. Table A shows radii 

 of curvature varying from 0.2977 to 5,000 or, 

 for the cemented siu-face alone, from 0.2977 

 to 0.4671. The second solution, or Table B, 

 shows radii for this middle surface of from 

 0.1705 to 0.3495. On this account therefore 

 Table A gives the more useful patterns. An 

 additional table gives for each type the 

 amoimt of coma left uncorrected, which 

 averages nearly the same for A as for B. 



Both A and B are calculated for the ar- 

 rangement of crown lens preceding, flint fol- 

 lowing. The reversed arrangement is pro- 

 vided for in Tables E and F, and these call 

 for radii which are individually and on the 

 average considerably smaller, curvature there- 

 fore greater; but in E, the coma remaining 

 in the system is somewhat reduced. Other 

 tables are for forms where three radii are 

 equal and the fourth surface nearly flat, so 

 that the cost of grinding might be lessened 

 even though the telescope would be less efii- 

 cient. These last are accompanied by an 

 exhibit of the residual amount of both 

 spherical aberration and coma. Two further 

 tables promise freedom from coma, with 

 stated amounts of uncorrected spherical aber- 

 ration. 



So far, it has been assumed that the thick- 

 ness of the lenses is so small as to be negli- 

 gible. Of course the diameter that is needed 

 for a particular pm-pose may cause a thick- 

 ness which is far from negligible, especially 

 in types having one or more fairly large cur- 

 vatures. To allow for this, the authors fix 

 arbitrarily a " standard thickness " of one- 

 fortieth the focal length for a convex lens, 

 one eightieth for a concave, and fiu-nish for 

 these standards thicknesses tables of two 

 sorts. The first shows how much the focal 

 length is diminished by standard thickness 

 when one uses the radii taken from a thin- 

 lens table, and the second shows by what 

 amount the curvature of the fourth surface 

 (the most nearly flat) may be modified to 

 restore the focal length to its intended value, 

 unity. 



Such an alteration of one surface is how- 

 ever only a make-shift, as is seen from the 

 later tables (1916), "Additional data," etc. 

 To alter the curvature of a single one of the 

 four surfaces disturbs not only the focal 

 leng-th, but also the precise balance of both 

 the aberrations which are already eliminated. 

 The authors recommend it indeed only when 

 the focal length is to be short. Otherwise it 

 is necessary to change slightly all three cur- 

 vatures from the 1915 tables. Very full in- 

 formation is given as to the amount of 

 change. First they give factors for inter- 

 polation when either index differs slightly 

 from that for which the earlier tables were 

 computed. Then back to this table are 

 referred, in the next following series, the 

 effects of standard thickness upon chroma- 

 tism. Namely, the corresponding change to 

 be made in the ratios of indices for flint and 

 crown is stated, so that by two tables the 

 changes of curvatures can be found. Next 

 comes the effect upon spherical aberration 

 resulting from standard thickness, and last, 

 the necessary changes in curvatures to correct 

 that error. But it is recommended that when 

 two kinds of aberration simultaneously be- 

 come serious in amount, the curvatures be 

 computed entirely de novo, since the errors 

 are not wholly independent. Such computa- 

 tion is of course greatly facilitated by knowl- 



