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final values of the unknowns a, b, c, d, e, f, &c., the following expressions:

a=1.

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52

72

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112

3a — 1o ' 5o — 1o · 7o — 1o ́ 92 — 1o ́ 1 12 – 1o

— · — · - · &c.,

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The quantity or a quarter of the circumference is equivalent, according to Wallis' Theorem, to

2.20 4.22

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7.7

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2.24 4.26

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If now in the values of a, b, c, d, &c., we notice what are the factors which must be joined on to numerators and denominators to complete the double series of odd and even numbers, we find that the factors to be supplied are:

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177. Thus the eliminations have been completely effected, and the coefficients a, b, c, d, &c., determined in the equation

1 = a cos y +b cos 3y+c cos 5y + d cos 7y + e cos 9y+&c. The substitution of these coefficients gives the following equation:

π

4

= cos y

1 3

1

1

cos 3y+cos 5y—cos 7y+cos 9y- &c.2

The second member is a function of y, which does not change in value when we give to the variable y a value included between - and +. It would be easy to prove that this series is always convergent, that is to say that writing instead of y any number whatever, and following the calculation of the coefficients, we approach more and more to a fixed value, so that the difference of this value from the sum of the calculated terms becomes less than any assignable magnitude. Without stopping for a proof,

1 It is a little better to deduce the value of b in a, of c in b, &c. [R. L. E.] 2 The coefficients a, b, c, &c., might be determined, according to the methods of Section vI., by multiplying both sides of the first equation by cos y, cos 3y, 1 1 cos 5y, &c., respectively, and integrating from to, as was done by 2 D. F. Gregory, Cambridge Mathematical Journal, Vol. 1. p. 106. [A. F.]

2

which the reader may supply, we remark that the fixed value which is continually approached is, if the value attributed to y is included between 0 and 17, but that it is -17, if y is included between and π; for, in this second interval, each term of the series changes in sign. In general the limit of the series is alternately positive and negative; in other respects, the convergence is not sufficiently rapid to produce an easy approximation, but it suffices for the truth of the equation.

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belongs to a line which, having x for abscissa and y for ordinate, is composed of separated straight lines, each of which is parallel to the axis, and equal to the circumference. These parallels are situated alternately above and below the axis, at the distance, and joined by perpendiculars which themselves make part of the line. To form an exact idea of the nature of this line, it must be supposed that the number of terms of the function

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has first a definite value. In the latter case the equation

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belongs to a curved line which passes alternately above and below the axis, cutting it every time that the abscissa a becomes equal to one of the quantities

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According as the number of terms of the equation increases, the curve in question tends more and more to coincidence with the preceding line, composed of parallel straight lines and of perpendicular lines; so that this line is the limit of the different curves which would be obtained by increasing successively the number of

terms.

SECTION III.

Remarks on these series.

179. We may look at the same equations from another point of view, and prove directly the equation

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The case where x is nothing is verified by Leibnitz' series,

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We shall next assume that the number of terms of the series

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instead of being infinite is finite and equal to m.

We shall con

sider the value of the finite series to be a function of x and m. We shall express this function by a series arranged according to negative powers of m; and it will be found that the value of the function approaches more nearly to being constant and independent of x, as the number m becomes greater.

Let y be the function required, which is given by the equation

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This equation

m, the number of terms, being supposed even.

differentiated with respect to a gives

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sin 2x 2 sin x sin 2x - 2 sin 3x sin 2x + 2 sin 5x sin 2x ...

=

+ 2 sin (2m - 3) x sin 2x - 2 sin (2m-1) x sin 2x.

10

Each term of the second member being replaced by the difference of two cosines, we conclude that

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cos (2m - 3x) + cos (2m + 1) x.

The second member reduces to

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1

or sec x

180. We shall integrate the second member by parts, distinguishing in the integral between the factor sin 2mx dx which must be integrated successively, and the factor which must be differentiated successively; denoting the results. of these differentiations by sec' x, sec" x, sec" x,... &c., we shall have

COS X

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which is a function of x and m, becomes expressed by an infinite series; and it is evident that the more the number m increases, the more the value of y tends to become constant. For this reason, when the number m is infinite, the function y has a definite value which is always the same, whatever be the positive

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