26 o 
Mr. Barlow on the rules and principles for 
focal lengths, being thus very exactly the same as required 
by the conditions of the problem. 
The lenses ground to these numbers turned out very fine ; 
the surface, centering. See. was also very perfect, and the 
result, notwithstanding the discrepances between the com- 
puted and actual radii employed, was very satisfactory. 
The scale of the experiment was however too small, the 
diameter of the lens being only 2^ inches, so that the defect 
of correction for aberration was not very sensible ; but in 
several other subsequent experiments, with very nearly the 
same proportional numbers for focal lengths of 5, 6, and 
7 feet, with proportional apertures, the want of correct 
balance in the spherical correction was very manifest. 
It seems therefore that we may in some cases deviate from 
the radii given by theory much more than in others, without 
producing the same defect in the instrument ; and it will be 
seen that this ought, a priori, to be expected. We know that 
the amount of aberration (the focal length, aperture, See. 
being given), varies with the ratio of the surfaces, and is 
least in the plate lens for all the usual indices for parallel 
rays, when r : r' : : 1 : 6 , and is very little increased with the 
ratio of 1 : 1 ; all those results therefore, which require a 
ratio comprised within, or near, these limits, will have but a 
small quantity of aberration in the plate lens to be corrected 
by the flint lens ; but when we employ such numbers as 
require a ratio of 3 to 1 , or 4 to 1 in the plate, then the aber- 
ration to be corrected by the flint is very considerable, and a 
small discrepance between the computed and practical radii 
will produce a much greater error than the same discrepances 
would in the former instance ; and to this circumstance I attri- 
