2. At what time between 2 and 3 o'clock will the hour and minute

hands of a clock be together?

In this case, the minute hand must evidently gain two revolutions, or 24 spaces. Hence, d=24; and we have by the formula,

3. What time between 2 and 3 o'clock will the hour and minute hands be at right-angles to each other?

In this case the minute hand must gain 24 revolutions; that is, d 12×2 27. Hence,

4. What time between 5 and 6 o'clock will the two hands of clock be in the same straight line?

Here the minute hand must gain 5 revolutions; and d = 12x 5 66.

512

Hence,

That is, the hands make a right line at 6 o'clock, a result manifestly true.

We will now apply this formula to certain motions of the heavenly bodies. It is known that the moon has a real motion around the earth from west to east. The sun also has an apparent motion in the same direction, in consequence of the real motion of the earth around the sun. The time of new moon is when the moon is in the direction of the sun from the earth, or when the moon is passing the sun, in her motion. With this explanation we present the following problem :

5. The average daily motion of the moon around the circle of the heavens is 13.1764°, and the apparent daily motion of the sun in the same direction is .98565°. Required the time from one new moon to another.

To apply the formula, we have

d = 360°, a = 13.1764°, b = .98565°, and a-b12.19075°. Hence,

6. The planet Venus, as seen from the sun, describes an arc of 1° 36' per day, and the earth, as seen from the same point, describes an arc of 59'. At what intervals of time will these two bodies come in a line with the sun and on the same side of it?

=

Here d 360° 21600', a = 1° 36', and b 59'. Hence, = = a-b = 37', and we have

The data in the last example were not taken with extreme accuracy, the object being mainly to illustrate a method. More exact data would have given 583.92 days.

INEQUALITIES.

195. An Inequality is an expression signifying that one quantity is greater, or less, than another; as

a>b, and c<d.

In every inequality, the part on the left of the sign is the first member, and the part on the right the second member.

196. In treating of inequalities, the terms greater and less, must be understood in their algebraic sense, which may be defined as follows:

Of any two quantities, as a and b, a is the greater when a—b is positive, and a is the less when a―b is negative.

197. From this definition it follows, that

Any negative quantity is less than zero; and of two negative quantities, the greater is the one which has the less number of units. Thus,-20, because -2-0-2, a negative result; and -3-5, because-3-(-5)=+2, a positive result.

198. Two inequalities are said to subsist in the same sense, when the first member is the greater in both, or the less in both. Thus a> d and c > d, or u < z and x <y, are inequalities which subsist in the same sense. But the inequalities, m>n and p < q, subsist in a contrary sense.

PROPERTIES OF INEQUALITIES.

199. Inequalities are frequently employed in mathematical investigations; and to facilitate their use, it is necessary to establish the following properties:

I. An inequality will continue in the same sense, if the same quantity be added to, or subtracted from, each member.

For, suppose

a > b.

Then according to (196), a-b is positive. Hence,

(a±c)—(b±c)

is positive, and consequently

a+c> b+c.

It follows obviously from the principle just established,

1.—That a term may be transposed from one member of an inequality to another, by changing its sign.

2. That if an equation be added to an inequality, member to member, or subtracted from it in like manner, the result will be an inequality subsisting in the same sense.

II. If an inequality be subtracted from an equation, member from member, the sign of inequality will be reversed.

III. If the signs of all the terms of an inequality be changed,

the sign of inequality will be reversed.

For to change the signs of all the terms is equivalent to subtracting each member from 0=0.

IV. If two or more inequalities subsisting in the same sense, be added, member to member, the resulting inequality will subsist in the same sense as the given inequalities.

For if

a> b, a' > b', a">b", .....,

then from (196),

a-b, a'-b', a'"—b", .

.....

are all positive; and the sum of these quantities,

a—b+a'—b'+a”—b", or (a+a'+a'')—(b+b'+b′′),

is therefore positive. Hence,

a+a+ab+b'+b".

It is evident that if one inequality be subtracted from another established in the same sense, the result will not always be an inequality subsisting in the same sense. Thus, it is evident that we may have

a> b and a' > b',

in which a―a' may be greater than b-b', less than b—b', or equal to b-b'.

V. If one inequality be subtracted from another established in a contrary sense, the result will be an inequality established in the same sense as the minuend.

For, if

and

a> b a' <b',

(1)

(2)

then ab is positive and a'-b' is negative; therefore, a-b(a'-b'), or its equal (a-a')-(b-b') must be positive, and we

shall have

a-a'>b-b',

an inequality subsisting in the same sense as (1).

If (1) be subtracted from (2), member from member, it can be shown, in like manner, that

a'—a <b'—b.

VI. An inequality will still subsist in the same sense, if both members be multiplied or divided by the same positive quantity.

1

m

For suppose m to be essentially positive, and

a> b.

Then since a- -b is positive, we shall have both m(a-b) and (a) positive. Therefore,

VII. If both members of an inequality be multiplied or divided by the same negative quantity, the sign of inequality will be reversed. For, to multiply or divide by a negative quantity will change the signs of all the terms, and consequently reverse the sign of inequality, (III).

VIII. If two inequalities subsisting in the same sense be multiplied together, member by member, the sign of inequality remains the same when more than two of the members are positive, but is reversed when more than two of the members are negative.

Products, aa' > bb'

aa' <bb' —aa' < bb' aa'>—bb'

The first two results are evident from the fact that when the two members of an inequality are both positive, the greater member has the greatest numerical value; but when the two members are both negative, the greater member has the least numerical value.

The other two results are evident from the fact that any positive quantity is greater than any negative quantity.

It will be found that if two of the four members are positive and two negative, the result will be indefinite.

REDUCTION OF INEQUALITIES.

200. The Reduction of an inequality consists in transforming it in such a manner that one member shall be the unknown quantity standing alone, and the other member a known expression. The inequality will then denote one limit of the unknown quantity.