n, o, p; from which points draw the lines IAF, m BG, nCH, o DI, þEK, as if they were rays of light that came from a focus R, and were reflected from the bafe lnp; fo that each couple as IA, IR produced may cut off equal fegments from the circle. Laftly transfer the lines laf, mbg, &c. and all their parts in the fame order, upon the respective lines IAF, m BG, &c. and having drawn regular curves, by eftimation, through the points A, B, C, D, E, and through F, G, H, I, K, and through every intermediate order of points; the figure ACEKHF fo divided will be the deformed copy of the fquare drawn and divided upon the original picture; and will appear fimilar to it when feen in the polifhed cylinder, placed upon the bafe Inp, by the eye put into its given place O. And the fame directions that were given in art. 637, ferve for painting the deformed copy of the picture upon the deformed fquare AK. For fuppofing the eye at O, raised to the height RO, to be a luminous point; every fuperficies of rays emitted from it towards every line of the fquare and confequently towards every line of its fhadow a k, being intercepted by the furface of the cylinder, will be reflected to every correfponding line of the deformation AK; as will eafily appear by comparing the folution of the lemmas and their corollaries with that of this problem. Therefore on the contrary, the rays which flow backwards from every line of the deformation AK, will be reflected to the eye as if they came directly from the lines of the shadow ak or from the lines of the fquare it felf. 2.E. D. 643. Corol. 1. If the ratio of the fides of the original picture, and con- Fig. 538. fequently of the fimilar parallelogram aex, be given; and R be given or affumed; and nt, the height of the cylinder, or of the highest point of reflection, be alfo given; the height of the eye may be determined by placing nt perpendicular to n R, and by placing the parallelogram aexupon its bafe ace at any convenient distance behind the arch lp, and by producing its base ca till c be equal to its height; then the line et produced will cut off the required height RO; as will appear by conceiving the triaugle ROteh to ftand perpendicular to the paper upon the line Rnch. And if the data be varied, the reft may be easily determined by the rela tion of the lines we have been confidering. 644. Corol. 2. If the portion of the cylindrical surface, which reflects the light, be extended into a plane, the deformation AK will become exactly equal to the fhadow ak. The 527, 528, and 529th figures belong to a concave cylindrical furface. Fig. 531,532. a Art, 211. Fig. 533. CHAPTER XIII. The theory of the aberrations of rays is refumed and carried farther, in order to difcover the limits of perfection of reflecting and refracting Microscopes; and to determine for them what was determined for Telescopes in the feventh chapter. 645. rays. H PROPOSITION I AVING the focus of homogeneal rays incident upon a Let 2 and q be conjugate focuses of incident and refracted or reflected n 0m2 2 2+= 8 2+ -s Draw BSZM parallel to 12, cutting IK produced in Z; and let SA be perpendicular to 21; and let t and M be the focuses of refracted rays, fuppofing the incident ones to have come parallel and the nearest to the femidiameters SO, SB; and by art. 224, SM=St=7S; and if the ray тв 27 was parallel to SO, then would its aberration ty=X by art. 329; and by art. 333 the aberration MZ:ty::SA square: IX square:: QS square * Art. 204. QI square or 20 square*. Hence MZ or Z=2S]2 X x= X; and SZ= 2-812 nn 22 тв S-Z. Now because the triangles QKI, SKZ are fimilar; it is as QK :: SK :: QI: SZ'; and conjointly as 2K: QS::21:21+SZ. Whence point I comes to O and K to q, the conjugate focus to 2, we have 2q= 2-5×2; and fo the aberration qK= (QK-Qq=)2−S× the arch IE cutting SO in E; and calling OE, E, we have 2-I-E and + because 2→Sis vastly greater than 2+7-S×ZE. Now by art. ration qK= 2. Now if the ray be reflected at I, for n substitute -m and confequently Fig. 532. 2 m for §, and the theorem changes into this, qK= I 2 XX. For the calculation is the fame whether the refracted ray goes backward or forward in the line IK; and to change the angle of refraction SIK into an angle of reflection, it (and its fine n) must be diminished to nothing, and then be made negative and equal to (-m the fine of) the angle of incidence S12. And during this change the method of calculation continues unaltered. 2. E. J. 646. Corol. 1. Put r for IX, and for a refracted ray the aberration qK 647. Corol. 2. Let S become infinite, and by the last corollary the ab mmnn rr mn 22 erration of a ray refracted at a plane furface, that is q K= and the aberration of a ray reflected at a plane furface is nothing at all. 648. Corol. 3. When the point of incidence is given, the longitudinal aberration qK of a reflected ray, from its focus q, is as Sq2, the fquare of the distance of that focus from the center of the furface. For by this propofition qK is to X, the aberration from 7 the focus of parallel rays, as a Art. 334. QS2 to 272, or as Sq2 to ST2; because QT, ST, qT, being continual proportionals, are proportionable to their fums or to their differences. b Art. 207. Fig. 534 649. Corol. 4. When and S are given the longitudinal aberration of the outmost reflected or refracted ray is as rr, the fquare of the femiaperLure of the furface, by corol. 1. LEMMA I. 650. When the terms of a ratio involve two forts of quantities, one of which is incomparably finaller than the other; the magnitude of that ratio is not altered by any alterations made in those infinitely fmall quantities. This will be evident when applyed. LEMMA II. 651. In the produced femidiameter or of the spherical surface o E, let and p be conjugate focuses of refracted rays; alfo x and k two other conjugate focuses; and when the interval x of the focuses of incident rays is exceeding small, the interval pk of the focuses of the refracted rays will Xax; fuppofing m, n, to fignify the fame as in the former be propofition. For lett be the focus of parallel rays coming the contrary way to the incident ones that flow from a, and by art. 224, ot=" or and rt=" or; and by art. 238, wt: wr::wo:p, and conjointly wt:tr::wo: po, that is (for or, ow, ot, op putting r, w,t,p)","r−w:] r :: w; p = ; and confequently putting k and x instead of p and w, we have k nra = nrx mr-ox *Art. 650. X@x*= Fig. 535,536. 2 Art. 211. ; and fo the distance pk=(p-k=) mnrr Xx, because the fquare of any quantity is the fame whether its root be affirmative or negative. 2. E. D. PROPOSITION II. 652. Having the focus of homogeneal rays incident upon any lens, it is propofed to find the aberrations of the refracted rays. Let OIEo be the given lens, whofe vertexes are O and o; R the center of the first surface OI;r the center of the fecond o E; and in the axis oOr R let P be the focus of incident rays, and p the geometrical focus of the refracted ones; it is propofed to find pl the aberration of the ray PIEL. From the points of incidence and emergence I, E, draw IV, Ev perpendicular to the axis, and putting D for the difference of the thickneffes Oo, Vv; P,R,r for the lines OP, OR, er; and 2 to 3 for the ratio of refraction in glass, I fay the the aberration pl= 6PPXR-r2-24 PRrx R-r-+24 RRrr fuppofing the lines P, R, r to lye all on one fide of the glass; and when they have different pofitions in different glaffes, their figns in the theorem must be altered accordingly. X 6PR For let be the conjugate focus to P after the first refraction at I, and Ix the first refracted ray; then calling VI and v E, I and E, and in cor. I. prop. 1. for 2, S, m, n, 8, Yput P, R, 3, 2, 1, I, and the first aberration P-RI 4P-10R XII. Again taking x IE for the incident ray P+2 R upon the second surface o E, let k be the conjugate focus to x after the second refraction; then call ox and O, x and; and in cor. I. prop. 1. for 2, S, m, n, 0, rput x, r, n, m, -8, Eor I, (for they are equal, being in the ratio of xv tox,) and the fecond aberration kl= r 2 mmrm nr — mm x II. But p is the conju-Art. 650. 6rr by putting t for P+ 8r3 × P-R × 2P — 5 R RP—ri|2 × 36RrP 5 II, and II. By adding pk and kl together and re ftoring the value of and dividing all by 3, 27 R3 pl= +33 RRr - + 7r3 PP +66 R3r 2 we have the total aberrration - 52 R2 r2 > P -52 R3rr ➡52R2 r3 12 Rr PP×R−r|3 −48 R Rrr P× R—r — 48 R3 r3 II. II 2R D. By putting this value inftead of II and dividing every term by R-r, we have |