Some New Properties in Conic Sections, Discovered by Edward Waring, M. A. Lucasian Professor of the Mathematics in the University of Cambridge, and F. R. S. to Charles Morton, M. D. Sec. R. S.

XXXV. Some New Properties in Conic Sections, discovered by Edward Waring, M. A. Lucasian Professor of the Mathematics in the University of Cambridge, and F R. S. to Charles Morton, M. D. Sec. R. S. THEOR. I. Read June 21, 1764. SIT ellipsis APBQCRDSET, &c. describantur circa eam duo polygona [Tab. XIII. Fig. 1.] (abcdef, &c. pqrstv, &c.) eundem laterum numerum habentia, &c quorum latera ad respectiva contactuum puncta (APBQCRDS, &c.) in duas æquales partes dividuntur, i.e. \(aA = Ab\), \(bB = Bc\), \(cC = Cd\), &c. \(pP = Pq\), \(qQ = Qr\), \(rR = Rs\), &c. &c erit summa quadratorum ex singulis unius polygoni lateribus æqualis summae quadratorum ex singulis alterius polygoni lateribus, i.e. \[ab^2 + bc^2 + cd^2 + de^2 + ef^2 + \ldots + \text{&c.} = pq^2 + qr^2 + rs^2 + st^2 + tv^2 + \ldots + \text{&c.}\] Cor. Ducantur lineæ AB, BC, CD, DE, EF, &c. PQ, QR, RS, ST, TV, &c. &c erit \[AB^2 + BC^2 + CD^2 + DE^2 + EF^2 + \ldots + \text{&c.} = PQ^2 + QR^2 + RS^2 + ST^2 + TV^2 + \ldots + \text{&c.}\] THEOR. II. Iisdem positis sit O centrum ellipseos, &c. ducantur lineae OA, OP, OB, OQ, OC, OR, OD, OS, &c. erit \[ OA^2 + OB^2 + OC^2 + OD^2 + \ldots = OP^2 + OQ^2 + OR^2 + OS^2 + \ldots \] Cor. Ducantur etiam lineae Oa, Op, Ob, Oq, Oc, Or, Od, Os, &c. &c. erit \[ Oa^2 + Ob^2 + Oc^2 + Od^2 + \ldots = Op^2 + Oq^2 + Or^2 + Os^2 + \ldots \] Hæc etiam vera sunt de polygonis inter conjugatas hyperbolas eodem modo descriptis. THEOR. III. Sit conica sectio MPQRSTM &c. [Fig. 2.] cujus diameter fit AL, et ejus ordinata ML; fit Mp = Mv, &c. consecuenter Lp = Lv. Ducantur lineae pq, qr, rs, st, tv, &c. quæ respectivæ tangant conicam sectionem in punctis P, Q, R, S, T, &c. &c. erit contentum \[ pP \times qQ \times rR \times sS \times \ldots = pq \times qr \times rs \times st \times tv \times \ldots \text{ vel, quod idem est, summa omnium hujus generis rationum } (Pp : Pq, Qq : Qr, Rr : Rs, Ss : St, &c.) \text{ erit nihilo æqualis.} \] Cor. 1. Sit ellipsis PQRS TV &c. circa eam describatur quodcunque polygonum (pqrs tuw, &c.), [Fig. [Fig. 3.] cujus latera respective tangant ellipsim in punctis P, Q, R, S, T, V, &c. &c erit contentum \[ pP \times qQ \times rR \times sS \times tT \times vV \times \text{&c.} = Pq \times Qr \times Rs \times St \times Tv \times Vw \times \text{&c.} \] Cor. Ducantur lineae PQ, QR, RS, ST, &c. &c pro finibus angulorum WPP, QPq, RRQ, QRR, SSR, TST, &c. scribantur respective a, p, b, q, c, r, d, s, &c. &c erit \[ abcd \text{&c.} = pqrs \text{&c.} \] Et sic de polygonis inter conjugatis hyperbolas inscriptis. Idem verum est de polygone, cujus laterum summa vel area minima fit, circa quamcunque ovalem in se semper concavam descripto, ut constat e nostra Miscell. Anal. **THEOR. IV.** Sit ellipsis PAQBRCSDTEVF, &c. [Fig. 4.] circa eam describantur duo polygona abcdef, &c. pqrsu, &c. eundem laterum numerum habentia; eorum latera ab, bc, cd, de, ef, &c. pq, qr, rs, st, tv, &c. respective tangant ellipsim in punctis A, B, C, D, E, F, &c. &c. P, Q, R, S, T, U, &c. &c sit \[ aA : Ab :: pP : Pq, \quad bB : Bc :: qQ : Qr \quad \text{&c.} \] \[ cC : Cd :: rR : Rs \quad \text{&c.} \] dD : De :: sS : St, &c. sic deinceps. Et area polygoni abcdef, &c. æqualis erit areae polygoni pqrsuv, &c. Cor. Duo parallelogramma (abcd &c. pqrs) circa datæ ellipseos conjugatas diametros (AC &c. BD; PR, QS) [Fig. 5.] descripta, erunt inter se æqualia. In hoc casu enim \(aA = Ab, bB = Bc, cC = Cd, dD = Da, \&c.\) \(pP = Pq, qQ = Qr, rR = Rs, sS = Sp; \&c.\) conseqüenter \(aA : Ab :: pP : Pq \&c.\) \(bB : Bc :: qQ : Qr, \&c.\) sic deinceps: ergo per theoremum hæc duo parallelogramma erunt inter se æqualia, quæ est notissima ellipseos proprietas. Idem dici potest de polygonis inter conjugatas hyperbolas eodem modo descriptis. **THEOR. V.** Rotetur conica sectio circa diametrum ejus (AL), \&c. sit \(MAM\), \&c. solidum exinde generatum; sint \(pq, qr, rs, st, tv, vw, wp, \&c.\) [Fig. 6.] lineæ, quæ tangant solidum in respectivis punctis \(P, Q, R, S, T, V, W, \&c.\). \& erit contentum \[pP \times qQ \times rR \times sS \times tT \times vV \times wW \times \&c. = Pq \times Qr \times Rs \times St \times Tv \times Vw \times \&c.\] **THEOR. VI.** Sit ellipsis \(APBQCR, \&c.\) rotetur circa diametrum ejus \(BD; \&c.\) circa conjugatas diametros (\(AC \& BD, PR \& QS\)) describantur elliptici cylindri (\(pqrs \& acbd\)) [Fig. 7.] solidum generatum circumscribentes, \& erunt hi duo cylindri inter se æquales. Sint duo solida e truncatis conis composita, solidum generatum circumscribentibus, \& quorum latera continuo tinuo eâdem ratione ad puncta contactuum dividuntur; erunt hæc duo solida inter se æqualia. Et sic de solidis inter conjugatas hyperboloides eodem modo descriptis. Facile constant plures consimiles conicarum sectio-num proprietates. Hujus generis proprietates affirmari possunt de infinitis aliis curvis, ut facile deduci potest e nostrâ Miscell. Anal. XXXVI. An