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and consequently fr sin" ——— cos x sin”-1 x—.

n

n-1

n (n-2)

cos r sin x

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cos r sin x —

n (n-2) (n-4)

COS 3

n (n- -2)

(n-1) (n-3) (n-5) n (n-2) (n-4) (n—6)

cos x sin &c....—

(n-1) (n- -3) .2 cos ; a formula which is true only when n n (n-2) (n-4)... 4

is an odd number, and then the integral depends simply on the quantities cos, sin x. But when n is even, instead of the last term of the series, which would be of the form—_(n—1) (n—3)

we shall have +

...

2 -2) 0 sin

...

1 cos x

(n—1) (n—3).....1 så sin”—” x=+ (n−1) (n−3).....1 2.4...(n-2) n

Consequently in this case the integral will be

2.4....n

I.

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88. Make r=90°—z, and we shall have *——, and sin x cos z

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5.3.1 6.4.2 Y.

89. Let it now be required to integrate y sin y cos "y; since flux

(sin 'y cos 1y) =p cos?+1 પુ sin-1 yy -q Cos?-1 y ein P+1

have fy sin-1 y cos? +1

y=

Therefore y sin "y cos "y=.

y y, we

p

sin y cos y+fy cos11y sin?+1 y.

P

1
m+1

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y. Substituting 1-cos y in place of sin 2y, and

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transposing, we have y sin "y cos "y=—

%-1

m+n

1

1

m+n

-fy sin" y cos"-2y=. sin+1 y cos1y+

m+n

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y cos"--5
(m+n) (m+n—2) (m+n−4)

sin+1

y cos"-3 ・y+. (n−1) (n−3) ... 2 sin”+1 (m+n) (m+n−2)...m+1

(n-1)(n-3) sinm+1

term will be.

y, if n is odd; or if n is even the last

(n-1) (n-3)... 1
Jy sin" y.
(m+n) (m+n—2) ...m+2

90. Make y=90°—z, we shall have ƒ cos" z sin"

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For example the first formula gives fy cos y sin' y sin y (cos y+) sin" y (-sin' y), and the second

Sy cos3 y sin3 y= cos y (sin +y+ sin 2y+3). Hence these two results ought to be equal, or at least to differ only by a constant

1

quantity. In the present case, this quantity is as we shall find

24

find by redncing the whole to sines, and comparing the two results.

91. Let us now consider those fractions in which sines enter;

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To integrate the second, let y=90°-z, and we shall have y=-i, and sin y=cos z; therefore

COS Z

- tan (45°-)-l cot (45°+ z)=7 tan (45° + z).

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92. This premised, let us investigate the integral of the formula

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We have already seen (87) that fy sin" y=

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fy sin-2y; therefore making n—2——m, or n=2—m we

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cos y (m-2) (m-4)

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COS sy

m-1 sinTM-1 y (m-1) (m−3) sin” y (m-1),(m-3)(m—5) sinTM3 y

...to the term + (m—2) (m—4)...1

y

; that is, to

(m-1) (m-5)...2 sin y

&c.

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93. Suppose y=90°-%, and the preceding formula will give

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For example,

y

cos'y

sin y+

5

cos y 6.4

sin y

5.3 +.

sin y

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5.3.1

+ Itan (45°+y).

costy 6.4.2 cosy 6.4.2

Hence it is easy to integrate the formula y cosm

odd number, as 2k+1, we have cos

sin" y

y, for if m is an

2k+1 y_flux (sin y)

sin" y

sin" y

(1-sing), which is evidently integrable for any value of n. If m

is any even number 2k, then y cos y_y (1—sin2 y)*

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which, when developed. is easily integrated by the formula for

95. The same process will also apply to y sin y; and the formula

y

Cos" y

may be integrated upon similar principles; so that it is sin" y cos" y

easy to integrate any fluxional expression containing sines and cosines, provided that they are susceptible of integration.

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ON THE INTEGRATION OF FLUXIONS CONTAINING SEVERAL VARIABLE QUANTITIES.

96. In any function t, of two variables x and y, if we first take the fluxion upon the supposition that x alone is variable, every term not originally containing some power of r will disappear, and the result will be of the form Pr. If we then take the fluxion of Pi, upon the supposition that y alone is variable, the result will be of the form P'ry, and by the double operation, all terms not originally containing both x and y, will have disappeared.

Again, if we take the fluxion of the same function 1, upon the supposition that y alone is variable, the result will be of the form Qy; and if we then take the fluxion of Qy, upon the supposition that alone is variable, the result will be of the form Qyr, all terms being excluded here also which did not originally contain some powers of both r and y.

Now in any term including powers of both x and y, we shall obviously obtain the same result, whether we take its fluxion, first supposing only a to vary, and then take the fluxion of the result, supposing y alone to vary; or whether we first suppose y alone to vary, and then x. Hence we shall always have Pry=Qyx, or more simply P'=Q'.

The quantities Px, Qy are called the partial fluxions of the function t, the first taken relative to r, the second relative to y; and P Q are called the partial fluxional coefficients.

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Example. Let the function be * +x3 y+a2 y2. Then the partial fluxion relative x is (4x3-3x2 y) x; and the partial fluxion of (4x3+3x2 y) * taken relative to y, is 3x2 xy.

.

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Again the partial fluxion of a+ + x3y + a2y2 relative to y is (x3 +2 a2 y) y, and the fiuxion of (x3+2a2 y) y, supposing & only to be variable, is 3x yx, the same result as before.

The partial fluxion of the function t,, taken relative to x, that is

supposing x alone to vary, is usually expressed by.

i

I

x; and the par

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