On Tube Length. By Edward M. Nelson. 129 



power in the list, it will be noticed that the percentage of error is not 

 large, Powell having got over the difficulty of the loss in magnifying 

 power owing to the shortening of the optical tube (see column F), by 

 increasing the power of the objective ; the lens is nominally a 4 in., 

 but in reality it is a 2^ in. In the second lens the nominal focus is 

 not given, so the values in K and L could not be filled in. 



Ross 3 in. is rather overdone. It has 2 in. more optical tube than 

 Powell's 4 in., and so the reduction in its focal length has been too 

 great. 



Powell's 2 in. (1876) is likewise overdone. 



Zeiss 24 mm. apo. is also overdone. A in this instance is very 

 nearly 10 in. 



Eoss 1 in. (1840) is another example of a lens with too short a 

 focal length ; his earlier objective, however, is about right. 



The Zeiss 12 mm. apo. is also overdone, the tube length being 

 slightly less than 10 in. ; and so to a greater extent are Powell's 

 ^ and l, the former being a ^ and the latter nearly a ^. Keichert's 

 8 mm. is an example in the other direction, the lens with a correct 

 tube length being considerably underdone. The Ross £ (1840) is 

 almost exactly right, both as regards focus and tube length. 



If the figures in column F and H are compared, it will be seen 

 in F that in those cases where the tube length A is less than 



10 in. there is a — sign in H, and vice versa ; but if a comparison 

 is made between the figures in E and H, a very different result will 

 be noticed; e.g., with a tube length d of 11*61 in E there is an 

 error of — 14 • 5 in H ; but with one of 12*8 there is an error 

 of 4- 2 • 7 ; another of 11*85 yields an error of 4- ' 8, and one of 



11 "51 gives 4- 4*1 in error. These results bear out the statement 

 above that the combined magnifying power is not proportional to the 

 tube length d. 



With regard to the formula for the initial power and the focus 

 in columns C and D, some explanation of their derivation is necessary. 

 Let a be the distance of the object from the lens, and a the distance 

 of the screen from the lens upon the other side. The relations of these 

 quantities to one another and to / the focus is fixed by the well- 

 known formula, which is given in every elementary text-book, viz. 



- = — 4- -. a and a are both positive as they are measured 

 fa a 



from the object to the lens, and from the lens to the image respec- 

 tively. There is one other common formula, viz. that the size of 

 the object bears the same proportion to the size of the image as 

 the distance of the object from the lens does to the distance of the 



image from the lens. Thus, if o is the object and i its image, - = 



—7 : but - is the magnifying power m, therefore m = — „ and a = — . 

 a o l a m 



April 17 th, 1901 K 



