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Rules and Principles for determining the dispersive Ratio of Glass ; 

 and for computing the Radii of Curvature for achromatic Object- 

 Glasses, submitted to the Test of Experiment. By Peter Barlow, 

 Esq. F.R.S. Mem. Imp. Ac. Petrop. #c. Read May 3, 1827. 

 [Phil. Trans. 1827, p. 231.] 



The author begins this paper by an enumeration of the various 

 works on the subject extant in our language, and a general mention 

 of the writings of foreign mathematicians, which he considers as 

 leaving room for further inquiry and simplification. He then states 

 the method employed in his experiments for determining the refrac- 

 tive and relative dispersive powers of his glasses, the former of which 

 is that generally known and practised ; of measuring the radii and 

 focal length of a lens, and thence deriving the refractive index ; with 

 some refinements in its practical application, consisting chiefly in 

 using the lens as the object-glass of a telescope, and adapting to it a 

 positive eye-piece and cross-wires, which are brought precisely to 

 the true focus by the criterion of the evanescence of parallax arising 

 from a motion of the eye, as is practised in adjusting the stops of 

 astronomical instruments. The only source of error it involves is 

 in the measurement of radii of the tools which it was found could 

 always be performed within i-^B-th of their whole values. The di- 

 spersive ratio of two glasses was determined by over-correcting the 

 dispersion of a convex lens of the less dispersive glass by a concave 

 of the greater, and then withdrawing the latter from the former till 

 the achromaticity is perfect, or as nearly so as the materials will ad- 

 mit, and measuring the interval between the lenses and their foci, 

 from which data the ratio of their dispersive powers is easily ob- 

 tained. 



The refractive indices and dispersive ratio thus determined, the 

 next step is to find the radii of curvature so as to destroy spherical 

 aberration. In this investigation, the author does not consider it as 

 necessary to limit the indeterminate problem by any further con- 

 dition, as others before him have done, but regarding it as a matter 

 of great convenience to avoid contact of the interior surfaces in the 

 centre of the glasses, leaves it open to the optician to make a choice 

 within certain limits, thus avoiding w r hat he considers as an intricate 

 equation arising out of the fourth condition. He proceeds, there- 

 fore, to express analytically the aberrations of the glasses, and to 

 deduce the equation expressive of its destruction, which of course 

 involves one indeterminate quantity ; this may be either of the radii, 

 or any combination of them. The author chooses the ratio of the 

 radii of the interior and exterior surfaces of his flint lens for this in- 

 determinate, which he assumes, as well as may be, to satisfy the 

 condition of the absence of contact and near equi-curvature of the 

 adjacent surfaces ; thence deduces, first, the radii of both of the sur- 

 faces of the flint lens ; next, its aberration to be corrected ; and 

 thence, by the solution of a quadratic, or by the use of a table con- 

 taining its solutions registered in various states of the data, the ratio 



