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And in the same way we may show that, if the corresponding terms of any number of proportions be multiplied together, the products will be proportional.

28. If three quantities are in continued proportion, the first has to the third the duplicate ratio of what it has to the second.

Let a, b, c be the given quantities in continued proportion; then

a 6 ūc

a ab aa a . Hence, or 7 x 5 = 7 X Ở = 723

a::: : b. And, similarly, if a, b, c, d are four quantities in continued proportion, a :d :: ao: 63, that is

The first has to the fourth the triplicate ratio of what it has to the second; and so on, for any number of quantities.

29. We shall now give one or two examples of problems in Proportion.

mam 03 c31a + C) 3 Ex. 1. Ifa:6::0:d, prove that 73 + di = 66 + d):

81

=

Let 7 = å = a; .:a = bx, and c = dx.

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(bx)3 + (da) — 68 + d®, then, alter

Hence,

(a

+ c)

(bac

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a + b _ Vac + Nöd Ex. 2. If a : 6 :: C:d, prove that y =

a - b Vac - Vod Let = = x; ."a = bx, and c = . nence, a 7 bu - 6 = x - 1 Hence, a + b = bx + b = x + 1 = Vbd.x + Vod

Tbd.x - Dod - Jox. dx + Jod Nac + Nod

Joc da - Nod Tac - Nod

Or, it may be worked thus

Since 5 – , we have Jo - Ja on, J = 1 = 4 x % or Hence, by Art. 26–

a + b Nac + vod

a - b Vac - Noď Ex. 3. If a : 6 :: c:d :: e:f, show that

ma" + nc" + peji

Imba + nd" + pfr) a Let =

......................(1).

Gio ova ::

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1. Zo di fr Hence, a= bxc", :: ma" = mb*z*)

c" = drz", nc" = nd och , and .:. by addition, e" = fræ*, :. pe= pf"x"

ma" + ne" + pe" = (mb" + nd" + pf)a". mas + nc + pea ormace + necas + per lo

) = ..(2). mbm + nd" + poft \mb+ ndá + pfr) :. Equating (1) and (2), we have a ma" + nca + pery

mbm + nd" + pf")

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).

Q.E.D.

Ex. VII: 1. Compare the ratios a + b:a b, and a2 + 6° : ao = 6*. 4. Compound the ratios 1 - 2c:1 +.y, 2 xy2 : 1 + ac?, and 1:3 - X".

2. Which is the greater of the ratios a + b : 2 a, and 26:a + 6?

3. What quantity must be subtracted from the consequent of the ratio a : 6 in order to make it equal to the ratio cid?

5. There are two numbers in the ratio of 6:7, but if 10 be added to each they are in the ratio of 8 : 9. Find the numbers.

6. In what cases is x + > 01 < 5 ?

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7. If

a b c 2 Y Ty

Wc show that a t b + c = 0. 8. Find the value of x when the ratio x + 2 a : x + 2b is the duplicate ratio of 2 x + a +0:2 x + b + c.

9. Find a when the ratio a - b : x + 2 a - b is the triplicate ratio of x - a : 0 + a - b. 10. If n te 80 + y y + 3C + 'a + 6 = 6 + c c + a' S

show that each of the frac

tions is equal to * # % + 6, and that = 3 = ,

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la t mc + nc , then each is equivalent to 16 to md. I nfi hence, show that

a
2 % + 2 – 3y 2 x + 2 y 2y + 2 % – a

When 2 a + 26-26 20 - a= 2 # 2 a - 6 12. If a :) :: c:d, then

a + b :c + d :: Va2 + ab + b* : Vca i cd. + d. 13. Find a fourth proportional to the quantities

2 + 1 2 + x + 1 2 + 1

x – I' m – 2 + 1 vnt - 1 14: Find c in terms of a and 6 when

(1.) a :a:: a - 6:6 - c.
(2.) a : 6::a 6:6 - c.
(3.) a :C::a - 6:6 - c.

15. If a, b, c are in continued proportion, show that a + b 67ab, bc are also in continued proportion. 16. If a:6::c:d, then

Lao " + do": N?" + d :: (a - b)* :(c - a)”. 17. From a vessel containing a cubic inches of hydrogen gas, b cubic inches are withdrawn, the vessel being filled up with oxygen at the same pressure. Show that if this operation be repeated n times successively, the quantity of hydro

(a - b)" gen remaining in the vessel is 'n-cubic inches, when reduced to the original pressure.

18. If, in Ex. 34, page 225, (ay, az, az), (61, 62, 63), and (C1, C2, C3) are corresponding terms respectively, show that exb3b3 (cz – cz) + a3b3b. (C3 – C1) + azb;ba (C1 – C2) = 0.

SEOTION III.

PLANE TRIGONOMETRY,

CHAPTER I.

MODES OF MEASURING ANGLES BY DEGREES AND GRADES.

1. We are able to determine geometrically a right angle, and it might therefore be taken as the unit of angular measurement. Practically, however, it is too large, and so we take a determinate part of a right angle as a standard.

In England we divide a right angle into 90 equal parts, called degrees, and we further subdivide a degree into 60 equal parts, called minutes, and again a minute into 60 equal parts, called seconds. This is the English or sexagesimal method.

In France the right angle is divided into 100 equal parts, called grades, a grade into a hundred equal parts, called minutes, and a minute into 100 equal parts, called seconds. This is the French or centesimal method, and its advantages are those of the metric system generally.

The symbols ,', ', are used to express English degrees, minutes, seconds respectively, and the symbols', ; ', to express French grades, minutes, seconds respectively.

Conversion of English and French Units. 2. Let D = the number of degrees in an angle, and G = the number of grades in the same angle; then as expresses the angle in terms of a right angle;

D

and so also does 100

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