Now rgT being the angle of incidence as above, and q1 the direction sin PQ T of the ray after one refraction, we have by optics, sin R F= and by trigonome in the triangle R Q F, sin n q Fsin itqF BF= it , and MF=R +1) ; also, representing the thickness at N of the lens by 1, ler R sin + 1) + S-t.
Then, by trigonometry, in the triangle n'44F, 8F+S-t WO get , sin Q F ain T'Q ; consequently by optics P +8 , 5 sin P'Q F = sin T'Ciri or the sine of the angle of refraction at the second surface, Now v'qP'-T'Qr+1./qe= (le m , or the angle which the second refracted ray makes with the axis of the lens : but by trigonometry, in the triangle R'qv', we have sin T''Q F sin sin log F ; whence Se = a }-;;-(-15-. -1)' Suppose next a double concave lens, the centres of whose surfaces are at re and n'", and whose radii are IV and a', to ho applied to the convex lens on the side N : then, neglecting the thickness of the con cave lens and the distance between the two, and supposing q F", (IF" to be the directions of the ray of light after the third and fourth refractions respectively, we have in the triangle li"Q F', by trigo. nomctry, 8' F' Bla Pit; F' = sin or the sine of incidence on the first surface of the second lens ; and by optics, n'+ s'F' -,- sin 1.'44 = sin T" Q F".
But r' Q V' - (T" - T"ct F")= QV' ; and in the triangle eq 1.'", by trigonometry, we have sin T"Q r" sin F" sin T Q wherefore Nr"=a' .n fogy." - -1); and considering Is a"' to be e equal to s', r" will be equal to N F" Again, in the triangle irq F", we have by trigonometry, sio n" q F"=sin gF"N for the sine of incidence on the fourth surface ; therefore, by optics, N F" -s' - Bill Q F" N = sin the sine of refraction at the fourth surface ; then Q F" N - v" 'q F'") = i"g P'", or =Q F"'N ; and by trigonometry, in the triangle Q r"' we have sin n'er'"sin no%no s' Ti,"Y, and ss.'"ee+sin or" s the focal distance of the compound lens.
Thaw values being reduced to what they become when the incident ray 11/ is Infinitely near the axis of the lennes that is, when the angles are substituted for their there may be Obtained R p 1 Si = = p NV IIF = d = moans of these equations, eliminating the quantities xv, ET', and Nr, and neglecting powera oft above the first, there may be obtained a value of ,r,,,: then differentiating this vague with respect to to p, p', and xr"', and making the resulting value of the differential of NT"' equal to zero (which is a condition necessary in order that the chro matic dispersion may be corrected for rays near the axis), there may be 2(p ) obtained a value of 2 + Again on substituting -- for n, and 1) for a, as above found, there will result p , ie ^1 I (1,tt= 4 NI + and r"' p I rip 1 1 + {1 + 1)t s' p 1 dp"Now the value of xr"' may be directly computed from the formulae first investigated; afterwards assuming different values of te, and sub stituting them in the last equation. let the corresponding values of
s' be found. With these values of s' find corresponding values of sin it'sor"' 8' sin or"'s + 1); that is, of xv'", and proceeding according to the usual methods of trial and error, there will at length be found a value of c r''' agreeing with that which was computed by the direct process : the four radii will then, consequently, be determined.
Investigations relating to the dispersion of light, and rules for com puting the radii of curvature for achromatic object-glasses, will also be found in an essay by Mr. P. Barlow of Woolwich, printed in the' Philo sophical Transactions' for 1827.
Though on thus uniting tho red and violet light by lenses of crown and flint sigma the chromatic dispersion is in a great measure corrected, yet when the image is examined, it is found to be surrounded by a green-coloured fringe. The difficulty of procuring flint glass of suffi cient purity is also A serious impediment to the perfection of achro matic lenses for telescopes. The steps that have from time to time been taken to remedy this evil are noticed in the following article on the history of the telescope, [Tetrecore, HISTORY or,) but we may here mention that in the' Transactions' of the Royal Society of Edin burgh, 1791, there Is given an account of some experiments, made by Dr. Blair, from which he was led to the discovery of the fluid medium, which, being applied between lenses of crown glass, renders the com pound lens completely achromatic. By adding liquid nniriatio acid to chloride of antimony, or sal ammoniac to chloride of mercury, he succeeded in obtaining a spectrum in which the coloured rays In each pencil followed the Name law of dispersion as takes place in crown gin". Therefore, confining a small quantity of either of these liquids between the convex surfaces' of two plano-convex lenses, or between those of a piano and a convex meniscus lens, of crown glass, Dr. Blair obtained an object-glAss In which the chromatic aberration was entirely destroyed; and he me maid to have thus constructed one of 9 Inches focal length, and as much as 3 inches in diameter or aperture. Objects ;dames so made were for some years on male in London; but either from the crystallisation of the fluids, or the negligence of the artists In compounding them, the telescopes became Imperfect, and gradually fell into disuse.