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of the former parts is then measured upon the small, or sliding portion, and that length similar to the unit is divided into n equal parts. It is evident that, if the sliding-scale is moved so that the first lines of division in each will coincide, the distance moved over will be equal to the difference between the length of the re1 1 (n-1)

spective parts of division, or,

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=

n

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Hence, if A B be an inch of such a scale divided decimally and C D be nine-tenths of an inch upon the sliding scale also divided into tenths: then, if C D be moved along till the position of the figures 1 and 1

1

1

coincide, the scale will have slid or ; if 2 and

2

102

2 coincide, then ; if 3 and 3, then

100

100

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The advantage of the application of the vernier is obvious, where accuracy and minuteness of admeasurement are necessary, as in the case of the barometer, in which the division above is the one generally adopted.

In its application to instruments for the measurement of angles the division usually adopted is that of 29 half degrees into 30 parts, when the coincidence of 1 2 &c. or,

the 1st and 2nd, &c. lines will indicate

1, 2, &c. minutes respectively.

-9

60 60

The same result is

obtained when 19 third-degrees are divided into 20 equal parts.

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1. Multiply a2
+(a−b) x -ab by

x + b.

Multiply x2

2. What is the principle of subtracting a quantity containing a minus term?

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c from a, show the working, and illus

trate it simply.

3. Subtract a2(b2-c2) from (a—b—c)2 and divide x3-3 a2 x- -2 a3 by x-2 a.

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1. Find the value of x in each of these equations—

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2. What number is that, the double of which exceeds its half by 6?

The difference of two numbers, and a quarter part of their sum are each equal to 2; find the numbers.

3. If A does a piece of work in 10 days, which A and B can do together in 7 days, how long would B take to do it alone?

Find the amount of P £ at compound interest for n years, the interest being paid yearly.

SECTION I.

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— a b by x + b.”

1. (1) "Multiply a2 - a x + x2 by
(2) "Multiply x2 + (a−b) x ·
(1)

a2

a x + x2

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2. (1) "What is the principle of subtracting a quantity containing a minus term?

(2) "Take b - c from a, show the working and illustrate it simply."

(1) Consider the sign positive and proceed as in addition.

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Here we have simply changed the signs of the quantities to be subtracted, and appended them to the a. Why the sign of b is changed is obvious; and why the minus sign of c is changed to plus is also easy of apprehension for when b has been subtracted by changing its sign, too much by the amount c has been taken from a; because it was not b which had to be subtracted, but b less by c. Therefore c must be added to

the result.

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· { a2— (b3—c2) }

(abc)2 = (a-b)-2 c (a—b) + c2

= a2 + b2 + c2 2 ab

a2 — (b2—c2) = a2 — b2 + c2

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2 ac+2bc

2ac2bc=2

2 (b-ab-ac + bc)

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2. (1) "Prove the rule for dividing one fractional quantity by another."

х

1

(2) "Divide (-) { + x + y }

x + y"

х

y

a

be a fraction which is to be divided by

by

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(1) Let

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But c was not the quantity by which we were to have

divided, it was the dth part of c.

Therefore the re

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