Before we proceed to the application of these transformations, it will be necessary to premise some properties of the functions λ and x'. -I COS. 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(x)) ; 9(9(9(x))) ; &c. we may most elegantly write p2(x); q3(x); &c. Thus we use d2x, ▲3x, Σ2x, for ddx, ▲▲▲x, 22x, &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 72 &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 p” q” (x) = q”+” (x). Hence if m = — n q” q—" (x) = 4° (x) = x. with the operation (9) performed no times on it, or merely x, that is, q―” (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 q′′x = q3”, q—”, x = 9′′ (9”, 4”, x), hence ", ", xx, and therefore 4" x = 4" 4" 9", x=4+", x. For instance d-n V="V, d'V=V. -Σ", x = "x.a" a, n times performed on it, and .. a" 2 an (cos.2 = x) &c.—and if c = 1 + = log.-2 I 1, with the operation of multiplying by 1, with the inverse operation so often per x= arc (sin. x). cos. 12 arc +12+ &c. 2 = log. c* and .. c* = log.—1 x. log.-"r, 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 contraction. wherefore cos. α= ^+^ and cos. «' = 2 2 Thus, λ=¿a√ ̄ ̄1 and x'=', where c = 1+ ÷ +÷2 + 1.2.3+&c. Hence, λ”+λ ̄”= 2. cos. n«, and x'”+x′ = 2. COS. na′; x" — λTM” — 2 √—1. cosin. na, λ”” — λ'-" = 2 √ — 1. sin. na'. Consequently, if k be any arc · k} = 4a". 2xa” . cos. k+λ12” = 4^^”. sin. (*+na') . sin. In like manner 1 2 2 We will now proceed to the application of these equations, and first, in Equation {1} for ◊ substitute, successively, each of a series of angles 4π 0 : 0 + 2 = 5 0 ; 0 + 4 = 0 ; ..... 0 + ; I n 2 n 3 And let the resulting values of r(1) be for the product of the several denominators of the (1) will be Ө x•} = 1— 2x” . cos. no +λ2 by COTES's theorem. n I This equation appears under an imaginary form when 471, 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 I 2 2 3 n I to each other; draw the axis ASM, and make the angle MSQn times MSP; and if L represent the latus rectum, we shall have for, by the polar equation of the curve, SQ=.c Thus, if SP be coincident with SA, and n be odd, cosec. but, if SP be perpendicular to SA, and n still odd, cosec. no If SP be perpendicular to SA, but n of the form, 4m + 2 Let us now resume the general equation {1,1} and first, let = 0, or, let one of the (1) terminate in the second vertex, This embraces all the cases where n is of the form (2m+1) , and among the rest, when n is any odd number, and one of the (1) terminates in the first vertex; when n is of the form 4m + 2, and one of the (1) perpendicular to the axis, &c. 3. Let cos. neo; then This includes the cases where n is of the form (2m + 1) . (7) as for instance, where one of the (1) is perpendicular to the axis and n odd, or, one inclined at an angle, and n of the form 4m+2, or, lastly, where 0 = and n=6m+ 3. {111}. This takes place whenever n is of the form (6m+1). and n = (6m + 1); 0 = , and n = 12m n is of the form 3m + 1. 2π 30 (3m + 1), and if 0 = 3′′, I We will now proceed to our second equation {2}, and by an operation exactly similar to that from which we obtained the equation {1,1}, we shall find 1 I-2e". cos. no+e3n I-2e". cos. no+e2n 1 -2. cos. ne 1 • {2,1} This transformation in «, however, as it possesses no particular elegance in point of form, and much complexity, we shall henceforward omit, except in a few remarkable instances. |