[8] II. On a remarkable Application of Cotes's Theorem. By J. F.W. Herschel, Esq. Communicated by W. Herschel, LL. D. F.R.S. Read November 12, 1812. LET a represent the semi-transverse axis of a conic section, ae the eccentricity, and consequently a (1— e2)=p the semiparameter. -e-2 2.(1) the distance between a point in the curve, and the focus, which, for distinction's sake, we shall call the first focus, and the adjacent vertex the first vertex: the others the second. 7(2) focus. the distance between the same point and the second Rits distance from the centre. its distance from the first vertex. 0 = the angle contained between the (1), and the prolongation of a line joining the first vertex and focus. the angle contained between the R and a line joining the first vertex and centre. the angle contained between the p and the same line. 0 is the angle whose supplement is, in physical astronomy, known by the name of " true anomaly," and is the corresponding" eccentric anomaly.” r(1) — a ( 1—e . cos. w); ¿(2) = a(1+e, cos. ∞) I-2A a2 (1—e2) (1+x2)2 I-2A. COS. 2a (1—e2) (1+x2)2. cos. ↓ COS. 1—2λ . cos. (%—4) + λa } ++^2 } { 1—2λ . cos. And lastly, since 1— e. cos. ☎ = · (1—e—1. cos. ✩) I-e. cos. w I-e2 cos. ' C we find cos. @= Before we proceed to the application of these transformations, it will be necessary to premise some properties of the functions λ and x'. but what is This notation cos. e must not be understood to signify usually written thus, arc (cos. e). It is true that many authors use cos. A, Sin.” A, &c. for (cos. A)", Sin. Am; lest therefore the notation here adopted should appear capricious, it will not be irrelevant to explain its grounds. If be the characteristic mark of an operation performed on any symbol, x, (x) may represent the result of that operation. Now to denote the repetition of the same operation, instead of 9 (9(x)) ; 9(9(9(x))) ; &c. we may most elegantly write p2(x); q3(x); &c. Thus we use d2x, ▲3x, Σ2x, för ddx, ▲▲▲x, ΣΣx, &c. By the same analogy, since sin. x, cos. x, tan. x, log. x, &c. are merely characteristic marks to signify certain algebraic operations performed on the symbol x, (such as &c.) we ought to write sin. 2x for sin. sin. x, log. 3x for log. log. log. x, and so on. To apply this to the inverse functions, we have q′′ q”. (x) = q′′+m (x). Hence if m = n p" 4" (x) = 4° (x) = x. with the operation (9) performed no times on it, or merely x, that is, ç―" (x) must be such a quantity that its nth (p) shall be x, or in other words "(x) must represent the nth inverse function. It frequently happens that a peculiar characteristic symbol is appropriated to the inverse function. Let it be, then ", x=4", x, and "x = p2", ", x = `p" (", ", x), hence ", ", xx, and therefore 4" x = ↓↓"q", x = 4+", x. For instance d-" V =ƒ»V, d°V = V . —Σ—”, x = A"x. a" = 1, with the operation of multiplying by a, n times performed on it, and .. a―" = 1, with the inverse operation so often per formed on it, ==; ·; α = ) Similarly sin. x = arc (sin. = x), cos. (cos. 2 = x) &c.—and if c=1+=+ = log.2x, and cc.....(n) x arc +&c. log. c and.. cx= log. x. 1 1.2 =log.―"x, or the nth inverse logarithm of x. It is easy to carry on this idea, and its application to many very difficult operations in the higher branches will evince that it is somewhat more than a mere arbitrary con traction. Thus, λ=c«v=i and 'c', where c=1+++12 n In n + 1.2.3+&c. Hence, λ”+λ ̃”—2 . cos. na, and λ“” +λ' ̄” n = 2. cos. na'; λ" — λ ̃” = 2√=1. cosin. na, λ 2 √1. sin. na. Consequently, if k be any arc 1— 2λ” . cos. k+λ12” — 4^"”. sin. (*+na') . sin. 2 2 We will now proceed to the application of these equations, and first, in Equation {1} for 8 substitute, successively, each of a series of angles for the product of the several denominators of the (1) will be 1 — 2x. cos. 0 + x2}... 2λ . cos. 0 + xa } { 1 — 2λ 1 {1—2λ. Ө n 2 cos. 0+x'} = 1—2x". cos. ne +22" by COTES's theorem. n λ This equation appears under an imaginary form when e71, -1 but, since cos. e is then a real angle, if we express it in a, it will then be free from imaginary symbols; thus When e = 1, or the conic section is a parabola, λ= 1, and A result of such remarkable simplicity, as deserves a more particular enunciation. Let then, in the diagram, fig. 1, S represent the focus of a parabola A,P,Q, and, having drawn any line SP, make n angles PSP, PSP .... PSP, about S, all equal I 1223 n I to each other; draw the axis ASM, and make the angle MSQ =n times MSP; and if L represent the latus rectum, we shall have I SP. SP . . . . . . SP = L"▬1 . SQ, 1 2 for, by the polar equation of the curve, SQ = .cosec. (MSQ). no Thus, if SP be coincident with SA, and ʼn be odd, cosec. ÷=1, and SP .... SP SP== L". n no but, if SP be perpendicular to SA, and ʼn still odd, cosec. If SP be perpendicular to SA, but n of the form, 4m + 2 |