When the force of gravity is 32.14, a pendulum 39.1 inches long vibrates in 1 second.

Compute the time of vibration of the same pendulum on the moon where the force of gravity is .165.

98. Velocity of Sound. The velocity of sound in gases varies directly as the square root of their elasticity and inversely as the square root of their density.

Sound travels through air at the rate of 1080 feet per second.

The elasticity of air is 1.6X105 and its relative density is 1.

FIG. 91.

Compute the velocity of sound in hydrogen whose relative density is .0694 and whose elasticity is 1.6X105.

99. Indicated Horse-Power. The coal consumed in tons by vessels of the same type varies jointly as the indicated horsepower and the time of passage in days.

The displacement in tons varies as the coal consumed.

Show that the indicated horse-power varies as the displacement, and inversely as the time of passage.

100. Initial Velocity of a Projectile. The square of the initial velocity of a projectile in feet per second varies directly as the charge of powder in pounds, and inversely as the weight of the projectile in pounds.

If 21 pounds of powder will give a 40-pound projectile an initial velocity of 2000 feet per second, compute the charge required to hurl a 50-pound projectile with an initial velocity of 1800 feet per second.

CHAPTER IV

THE FOUR FUNDAMENTAL OPERATIONS

SECTON 1, ADDITION. SECTION 2, SUBTRACTION.

MULTIPLICATION. SECTION 4, DIVISION.

SECTION 3,

59. Classification. The four fundamental operations are:

Addition,
Subtraction,
Multiplication,
Division.

They are called fundamental because they are basal in mathematics and other seemingly different operations are only these in various combinations. Their importance and the necessity of speed and accuracy in their application are therefore obvious.

60. Kinds of Terms. The terms of an algebraic expression are of two kinds, like and unlike.

Like terms are those having one or more letters the same with the same exponents.

Unlike terms are those which do not have the same letters with the same exponents.

What is meant by like terms can best be shown by an illustration:

ax, bx, rx, 5x, cx, and 2x,

are like terms with respect to x, all having the same letter x, affected by the same exponent 1.

5c2, 4a2c2, 10b3c2, and ac2,

are like terms with respect to c2.

Therefore to be like, terms must have one or more letters the same with the same exponents,

ac and bc2, 5c3, and 8c4,

although having the same letter c, are unlike because the exponents of c are unlike.

§ 1. ADDITION

61. Definition. Addition is the process of finding the sum of the terms of an expression.

To perform the addition it is convenient to write like terms in the same vertical column, making as many columns as there are unlike terms. The excess of plus or minus in each column is then determined and the result is written underneath.

62. Illustration. Suppose the following polynomials * are to be added:

3x+5x3-6x2+4x-1; 6x2-9x4+x3-13x+1;

12x3−8x2+16; and 6x4-11x-12x3 — 8.

Writing like terms in the same vertical column and finding the excess of plus or minus in each column, we have

* A polynomial is an algebraic expression of more than two terms.

63. Terms with Literal Coefficients. Like terms with literal coefficients or with both literal and numerical are added by inclosing the coefficients in a parenthesis followed or preceded by the letter with respect to which the terms were classified as like terms.

For example, if the sum of 5x-ax+cx-b2x is desired, the result is written (5-a+c-b2)x.

64. Law of Addition.

(1) Write like terms in the same vertical column.* (2) Find the excess of plus or minus in each column.

65. Examples in Addition. Find the sum of the following as specified by the law:

1. 3x-6x+2x4-8x3,

-12x2+11x-16x3.

3x3-6x+2x2-x,

7x4+9x3-x2+10x,

2. y1+3y3−2y2+5y, Sy2-7y+11y3-16y1, −y3+2y1+10y−8y2,

7y+15y-y2+y.

3. 422-6z+1223-2024+8, 62-1123-422+624-9,

224+323+9x2-30z+1, 8z3-7z+924-322+6.

4. -34t+18t2-714+8t3 -9, 7-16t2+t-8t3+9t4.

11t2-9t+14-7t3+3t,

5. ax+bx+cx+dx+ex.

6. 5x+3x-6x+cx+dx.

19t-t3+4t1-16+t2.

7. 6a-5b+7a-d, 3b-a+d-c, d+2c+a-2b, c-4d+9a-1, 1+8d+c-7a.

8. 5x2-4x2 bx2+3x+8x-x+cx.

9. a+5, 3a+8, -5a-7, -11a+16.

10. by+cy+dy+8y.

11. 4y2+6y+cy2.

12. 8x2+ax2+6x2-bx2-10x2+8.

13. 3x2-cx2+16x2 -tx2-4x2+7.

*

Usually unnecessary, since in algebraic work the addition of like terms can be determined from the expression as written.

14. 5x2+dx2-kx2-3x2+7x-3x+ax. 15. bx+cx-8x+9x2+fx-cx+3x2+x.

16. a2x+9x2-b2x — ax2+5x-dx.
17. 3√x−5√x+a√x_s√x.
18. 8√/bc+√bc-abc-3√/bc.

19. 5x2y-ax2y+5bx2y-2x2y+4cx2y.
20. ax2+3x2-x4+6x, 3x-bx2+x3+2x1.
21. (a+b)√x-c√x+d√x−5√x.

22. 3(x+y)-3x2+4(x+y)+16x2-c(x+y).

23. 4√b+c-8y+6-2√/b+c+15y+b√b+c-5,

26. 3xy-8x2y3+x′y$+7x′y3+34x2y3,

-bxy+12x2y3 —gx3y3+tx2y3 —y-1.

27. 24b2c3x2-30ax2+12by3+16b2c3x2,

-9by3+15b2c3x2+33ax2+21by3 — dax2.

28. rx2-dx2+5x2-tx2+3x2 — sx2+dx2 − x2.
29. 5x2+3x2-(a+b)x2+10x2-ax+5x+cx.
30. c4x2-b3x2-hx2+5x-3x+kx+8x2-5x+18.
31. 24.8x2-16.29x+7.05x+24,

-16+3.17x-32.2x2+1.48x3-18.3b,
17.6x2+12.38x3+16.7b-73.52x3-40.17.

32. a3+3ab2c-7a2bc, a2bc-3ab2c+a3, 15a2bc-8ab2c.

1 4

3

5

16

16

33. by − cy2+1ax3, ay+cy2 — bx3, dy+

34. r3x+s2y-tr2x−ay+c2y.