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By giving particular values to l, m, n ... an indefinite number of special relations may be obtained.

The relations of I. and II. are frequently employed with great advantage, the letters a, b, c, etc., being general symbols denoting any quantities or algebraic expressions.

Ex. 1. To find x from the equation (+2)

x

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α

ab

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Exercises of this kind may be solved in several ways: as (1) by transforming one expression into the other; (2) by using the first relation to show that the second is an identity, etc.

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Substitute for a and c in the second expression, and it becomes

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values of x, y, and z in terms of the remaining letters.

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whence, squaring each fraction and employing (5),

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Thus

Question 4 furnishes an example of collateral symmetry between the two sets of letters a, b, c and l, m, n. if we change a to b and b to c, we must, at the same time, change to m and m to n. But we are never supposed to make an interchange between letters from different sets. In like manner we may have collateral symmetry amongst three or even more sets of different letters.

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3. If

b

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i.

(a − c )2 + (b − d)2

=

810

α

b

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d

с

ii.

(a — c)2 − (b − d)2

a3 + a2b + ab2 + b3

a2 + b2

c2

b3

=

c3 + c2d + cd2 + d3 d3

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d2

then a√A+bB_a√A-b√B

c√C+d√D c√C-d√ D

(a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2.

4. Under the conditions of Ex. 3, show that

√(ab)+√(bc)+ √(cd)=√{(a + b + c) (b+c+d)}.

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CHAPTER VI.

RATIO, PROPORTION, VARIATION, OR GENERALIZED PROPORTION.

79. The ratio of a to b is the quotient arising from dividing a by b, where a and b denote any numerical quantities. If the division is even, the ratio is an integer, and is expressible; if uneven, the ratio is a fraction and can only be indicated.

In this relation a and b are called the terms of the ratio, a being the antecedent and b the consequent. The ratio is commonly symbolized as a : b.

If ab, the ratio is one of greater inequality.

If a= b, it is one of equality; and if a <b, it is one of less inequality.

When two ratios are multiplied together, after the manner of fractions, they are said to be compounded.

Thus ac bd is compounded of a: b and c d.

When a ratio is compounded with itself, the terms are squared, and the result is the duplicate ratio of the original. Thus a2: b2 is the duplicate of a: b. Similarly, a3 3 is the triplicate of a: b; and a3: √b3 is, in physics, sometimes called the sesquiplicate ratio of a: b.

:

80. As a ratio is virtually a fraction, all the laws of transformation for fractions apply to ratios.

The ratio of one quantity to another does not depend

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