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R......R=a". (1-72)" (1—^^) (1-23")

7,10}; but, if n = 6 m +1, R......R = "

(1-12)
V(1+12" +247)

{7,11}. These are always imaginary expressions when e 71 and n odd. In fact, R, in the hyperbola, must be written a ve

a vi-e?
instead of
Ne, cus. 02.

VI-e. cos. pa now ve-1e. Thus, this expression, like the rest, is easily transformed, in functions of a, a, a, the x' is now real, and the part involving a will always be of the form f (a" +1+),

+ 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: sin. (A + B). sin. (A – B) = 2

- )

P.Q, where า
A+

27 +(A+B) P=si'». (4+5). sin. (*+(A+). sin. A+B) sin. (n=1) (A+B)

n-*+) Q=sin. (4+*). sin. (*+(4=»). sin. :

-B –B) sin. ((n-1)+(A–B)

n(-) The demonstration is extremely simple, 1- 2.x", cos. a + x = (1 – 2x.cos. + x) (1 – 2x.

+ x?)...... (1 or dividing by **

x 2. cos, a = (x + x-'- 2. ) ....

2n2 2

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T

n

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n

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(a)

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2X

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a+2 (n-1)* + x>),

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n

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2

COS.

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nI COS, Q = 2

COS.

COS. nc

+

COS.

n

пс

at

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2n - 2 2

2

2

2

a+

.

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(r+x-'- 2.c a+2(n=1)").

Let x + x = 2. cos. C; then x" + x = 2 . cos, nc, and

(cos. C -
.
com) ...(

(cos. C – a+2 (n=1)"), that is (by the formula cos. — cos.y=- 2

> sin. My sin. $72). sin. 144mc). sin. *. sin. ( +c). sin. 1 (0*2* + c). &c. į

+) sin. 1 * - c). sin.

c). &c. Let #mc

= A + B, and 4=* = A – B, and by substitution, the formula under consideration results.

This immediately gives the following cos. (A + B). cos. (A – B) -?.P. Q, where () - (A+B) {: (1) – (A+)

} P = sin. (2n-1). (7) (7) – (A+B)

>(6) B) Q=sin.

2n-1) (*) – (A–B)
and also,
cos. (A + B). sin. (A — B) = 2 P.Q, where

{)
6) – (A+B) )

{ (*– (A+

3 6 ) – (A+B) P = sin. sin.

) (2n-1) () - (A+B)

(1– (A+ Q=sin. (4:3). sin. (*+(A=B) .... sin. (21)(A–B).

sin.

n

n

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NI

A

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sin.

n

n

... sin. (4+1/3+4. ()

n

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2

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=

.

21

2n

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Let B= 0, and (a) gives

25+A sin. A=2"–. sin. . sin.

-)+A

( If A= , this becomes 1=2"-'. sin. p. sin. 3. sin.

(2n-1) sin.

(e) (6) gives by making B= 0,

-A 3 (6)-A
)

(2n-1)
-116)-A

7 cos. A=2"-1.sin. ..sin.

sin.

--() , this gives 1=2". sin. T. sin. 7m. sin." sin.

sin. (615)-. (8) Equation (c) divided by (b) gives, (putting A, for A - B).

(N-1)+A tan. A

(h) }

(2n=117)But, to return from this digression, let us take Equation {8}, and putting it into this form

2p. (1+12). sin. 63.-)

(1-27.cos. +23) (1-23.cos. (7-4) +az) for substitute each of a series of angles, n in number

(-1) ;y ;; and let

be the resulting values of p.

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The values of sin. ( - ) in an inverted order, are (if

$ {4+ **** (*)}) sin. v; sin. (v + :); sin. (v + :);

. ..sin. (v + 7 + ), and their product, " TI, = sin.

sin. (2+) (by Equation (d)) = =

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(-1)

n

1

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4 + 4 + (+1)

¥

n

n

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2

(n-1)

n

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Again, the values of being 4,... \ + " ", those of 5 +$, or 27

(-) will be

4
Inti

(21-1)

中+ Now, cos. 27

(-4) = cos. (7-4). Hence, the product of all the denominators of p, p, &c. will be (1-21. cos. +

&– 22. 2.)(1—27. cos. (4+ 7) +19).... (1—21.cos. (4+

1)-) +2+). cos. (+ + 7) +4°).... (1– 22. cos. (4 + (2015) +20)=, by Cotes's formula

cos. 2n $ + **.
Thus we have, combining these separate processes,

sin. n E ++
= *
2a" (1 —4")". (1+x)**. 1–2X**.cos. zný + ***

{ } (1 +22)". sin. n (+++ (1–22]" . sin. n(+4)

2n-)

n

1

1

211 αλ

с

n

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1. Let n be any number of the form 4m + 1; then

11+12) 29. cos. n4

2p". P

{8,2}. 1-2221 cos. 2n +2411 If now p=0, or one of the p coincide with the transverse axis, 2p" (1 +12)an

(1-2221 . P (1-722

{8,3}

(1-22) Let, next, y * {n continuing = 4m +1} and, ?

1 +241

{8,4} In the parabola, 1 = 1. Hence in this case,

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P
I

•p

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= + (2L)".VI...

{8.5} In fig. 2, let ASM be the axis of a parabola, BAC a tangent at the vertex, S the focus—bisect the angle BAM by AP, and draw 4m other lines AP, AP, ..... AP, so that if

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2

PA, PA, &c. be produced through A, this system of lines shall make equal angles around A, then, neglecting the sign AP. AP ..... APE (2L)".VI

.

=

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I

n

If n = 4m + 3, the resulting value of p... pis of the same form as {8,2} with the exception of a different sign. If n be of the form 4m, or 4m + 2, +

{8,6},

n

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1

where the sign +,or --, is to be used, according as n is of the
former, or latter form.
When n. = 4m + 2, if ý = , this becomes

-
+ 2p".

{8,7},

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and when = 1.

(2L)*
P

P
=+
+ "

{8,8}. Thus, in the last mentioned construction, if the number of lines be 4m + 2, insiead of

4m + 1 AP, AP ..... AP

(2L)"

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The ninth of our primitive equations gives, if the angles

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