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 refractive 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 1/500th of their whole values. The dispersive 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 admit, and measuring the interval between the lenses and their foci, from which data the ratio of their dispersive powers is easily obtained. 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 condition, 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 what he considers as an intricate equation arising out of the fourth condition. He proceeds, therefore, 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 indeterminate, 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 surfaces 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 containing its solutions registered in various states of the data, the ratio of the radii of the convex, whence the radii themselves are easily deduced.
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