Correction of Telescopic Objectives. 411 



I now turn to an entirely distinct question — the reliability 

 of the results o£ the calculations and their subsequent treat- 

 ment. Is the solution of the algebraic equations simply an 

 equivalent of " experience " in affording a favourable basis 

 on which subsequent trigonometrical work is grounded, or is 

 it more ? On this score Mr. Allen is very decided : " both 

 the tables and the equivalent calculations lead to figures 

 such as no manufacturer with a reputation to keep up would 

 employ." 1 am bound to differ from Mr. Allen. So far as 

 my experience goes a manufacturer's reputation will be quite 

 safe if he solves the algebraic equations (not graphically, 

 since the solution will not be sufficiently accurate) for a thin 

 cemented objective, inserts the necessary thicknesses without 

 altering the curvatures found for the surfaces, and leaves the 

 objective as it is without troubling about trigonometrical 

 calculations*. Naturally this only holds within limits, 

 and may fail for abnormal combinations of glasses or for 

 abnormal apertures. It applies, however, to the general run 

 of objectives which are required in large numbers. In cases 

 where this procedure does not yield the particular type of 

 correction which the maker finds most pleasing, a slight 

 alteration should be made in the conditions imposed on the 

 thin objective. 



The low esteem in which Mr. Allen holds the algebraic 

 solution can hardly occasion surprise in view of a subsequent 

 statement. In obtaining his algebraic expressions he says 

 it is assumed " that all the angles in the calculation are.so 

 small that the excess of any angle above its sine is exactly 

 equal to a sixth of the cube of the angle. In other words 

 the rays could all travel within a capillary tube lying along 

 the axis of the lens.'" In saying " exactly " a somewhat 

 unhappy word has been chosen. To give a meaning to the 

 statement we may consider that what is meant is that, when 

 a definite number of figures are retained, the error resulting 

 from the neglect of the next term would only involve an 

 alteration of the last decimal place by unity, or alternatively 

 would just fail to alter it, the error not exceeding five units 

 in the next decimal place. Let four and five figures be 

 taken as illustrations since these are the number of figures 

 used in the majority of optical calculations by trigono- 

 metrical methods. The subjoined table gives the angles for 

 which the errors due to the neglect of (a) the second term in 

 the cosine expansion, (b) the second term in the sine expan- 

 sion, (c) the third term in the cosine expansion, and (d) the 



* For the theory underlying this use of the algebraic solution of the 

 first order conditions see Proc. Phys. Soc. vol. xxx. p. 119. 



