If then, n be even, this is equal to unity, as it evidently ought, We come now to our 7th Equation, which will afford us results, more complicated indeed, yet equally interesting. By applying the same method of transformation to it, we shall find, (supposing 4, 4, ...... 4 = 4 + 2 (n−1), to be written for n 9, and R, R, ..... R to denote the resulting values of R) 1. If n be even, cos. no cos. n (+4) and, since 1 — e 2. If n be odd, cos. no = — cos. n (# + q), whence Let = 0, or, let the extremity of one of the R lie in the Again, let one only of the R be perpendicular to the axis, and Next, let one of the R be inclined at an angle, to the axis. If n = 4m + 2, and it is curious to observe, that this expression is the same function of a, λ, n, as that of {7,7}. If n be of the form 2m + 1, Lastly, let n be of the form 6m + 2, and 4 = then Φ D & These are always imaginary expressions when e 7 1 and n odd. In fact, R, in the hyperbola, must be written m -m is easily transformed, in functions of a, a, x', the x' is now real, and the part involving a will always be of the form f (x" ±λ ̄m), and therefore readily expressed in trigonometrical functions. Before we proceed farther, it will be necessary to premise a transformation of COTES's formula, which we shall have occasion to make use of. It is as follows: P = sin. (^+B). sin. (*+(A+B)). sin. (2%+(A+B)) ..... sin. n A-B n Q= sin. (^—3) . sin. (*+(A—3)) . sin. n (a) a 2 (n−1) π ....... + x2), cos. a+2 + x2).... n COS. 2. cos. nc, and (cos. c 1+ (n-1)), that is (by the formula cos. — cos. y — — 2 . ž Let a+nc 2 = a-nc n A + B, and a—*c = A — B, and by substitution, 2 the formula under consideration results. This immediately gives the following cos. (A+B). cos. (A-B) = 22"-2. P. Q, where and also, n } . sin. { cos. (A+B). sin. (A — B) = 22-2. P. Q, where Equation (c) divided by (b) gives, (putting A, for A — B). But, to return from this digression, let us take Equation {8}, and putting it into this form P (1—2a. cos. ↓+λ3) (1—2λ . cos. (x−↓) +x2) for substitute each of a series of angles, n in number The values of sin. (—) in an inverted order, are (if 2 + "=2. (~)}) sin. v; sin. (v + 7); sin. (v + 2); .... sin. (v +7), and their product, = sin.. sin. n T ; sin. n(+4) 2 |