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98. One quantity is said to vary inversely as the former cannot be increased, but the other the same ratio; or the former cannot be decreas must necessarily be increased in the same ra former cannot be changed, but the reciproca changed in the same ratio.

EXAMPLE. A man walks a certain distanc if he walk twice as fast, he will go the giv an hour; but if he walk only half as fa require two hours to complete his journey of walking is inversely as the time he take

99. The sign ∞ placed between two q they vary as each other.

Thus A & B implies that A varies

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100. One quantity is said to vary a the former being changed, the produc necessarily be changed in the same 1.

Thus A varies as B and C join A cannot be changed into a, but the into bc, or that A: a:: BC: bc.

101. In like manner one quan jointly, when the former being " three latter is changed in the sam Thus ABCD, and the mconcerned.

EXAMPLE. The interest of I the principal, rate per cent, and

102. One quantity is said to inversely as a third, when the second multiplied by the reci second divided by the third,)

Thus A varies directly

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Cc, that is, AB & C. her, it will likewise vary as

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and a, b, c and d, corresponding values of them; and let A ∞ B, B∞ C, and Coc D; then will A ∞ D.

Because A: a :: B: b.

And Bb:: C: c.

And C:c:: D: d, therefore ex æquo (Art. 81.) A: a:: D: d, that is, A ∞ D; and the same may be shewn to be true of any number of variable quantities.

106. If the first be as the second, and the second inversely as the third, then is the first inversely as the third.

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107. If each of two quantities vary as a third, then will both their sum and difference, and also the square root of their product, vary as the third.

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Let A x C, and B∞ C, then will A+B ∞ C, and ✅AB

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Therefore ex æquali A:a:: B: b, and alternando A: B:: a: b, wherefore componendo et dividendo A+B: B :: a+b:b, whence alternando A+B: a+b:: B: b; but B:b : : C : c, wherefore ex æquali A+B : a+b:: C: c, that is, A+B ∞ C, or the sum and the difference of A and B will each be as C.

Again, because A :

And B :

Therefore (Art. 78.) AB :

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a: C: c,

b :: C : C,

ab :: C2: c2,

Whence (Art. 80.) √AB: √ab :: C : c, that is, ✅AB ∞ C. 103. If one quantity vary as another, it will likewise vary as any multiple or part of the other.

Let m be any constant quantity, and let A ∞ B, then will

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B b

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: :: mB: mb, Art. 73.

Therefore A a :: mB: mb, that is, A ∞ mB.

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Since A & B, A is equal to B multiplied or divided by some

constant quantity; for A: a :: mB : mb ::

nando A: mB::a: mb ::

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B b

: whence alter

m m

if m be assumed, so that A=mB, or

b

then will amb, or a=— respectively.

m

110. If the corresponding values of A and B be known, then will the value of the constant quantity m be likewise known. For if a and b be the known corresponding values of A and

B

B, then since A=mB, or A=—, by substituting a and b for A

m

b

and B, we shall have a=mb, or a=—

b

m

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› b

=: wherefore also (since A=mB, or A==) A==— × B, or=

a

m

X B.

a

111. If the product of two quantities be constant, then will the factors be inversely as each other.

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Let AB be a constant quantity, then is A œ for AB being constant, it may be considered as 1; that is, AB x

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112. Hence, in the constant product ABC, A ∞

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may be shewn when the product consists of any number of factors.

113. If the quotient of two quantities be constant, then are those quantities directly as each other.

Let

and B

A

B

∞ 1, then, (multiplying both sides by B,) will A ∞ B,

A, and the like may be shewn when the quotient is composed of any number of quantities.

114. If two quantities vary as each other, their like multiples and also their like parts will vary as each other respectively. Let A & B, and let m be any quantity constant or variable, A B

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For since by hypothesis A: a :: B : b, therefore mA : ma :: mB: mb (Art. 73.) that is, mA ∞ mB.

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115. If two quantities vary as each other, their like powers and like roots will vary as each other respectively.

Let A & B, then since A:a:: B : b (Art. 95.) A" : a" : : B" : b", and A : a :: B7 : b7, (Art. 79.) that is, Aa ∞ B", and

m

m

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n

m

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116. If one quantity vary as two others jointly, then will each of the latter vary as the first directly, and as the other inversely.

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For since BC ∞ A, divide both by C, and B OC

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117. If the first of four quantities vary as the second, and the third as the fourth, then will the product of the first and

third vary as the product of the third and fourth.

Let A & B, and C ∞ D, then is AC ∞ BD.

For A: a:: B: b.

And Cc: D: d.

Therefore (Art. 79.) AC: ac :: BD : bd; or AC ∞ BD. 118. If four quantities be proportionals, and one or two of them be constant, to determine how the others vary.

:

:

Let A B C D, then will AD=BC, and therefore AD ∞ BC. Let A be constant, then D c: BC, (Art. 104.) let D

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