If the density is given, then the weight varies as the length the breadth × the depth.

If the depth also is given, then the weight varies as the length X the breadth.

If the breadth is given, then the weight varies as the length. Finally, if the length also is given, then the weight is equal to a constant quantity.

(232.) One quantity varies inversely as another when one increases in the same ratio that the other diminishes. Thus, the altitude of a triangle whose area is given, varies inversely as its base.

If the product of two quantities is constant, then one varies inversely as the other.

In uniform motion, the space is measured by the product of the time by the velocity; that is,

If the space be supposed to remain constant, then

1 T∞

that is, the time required to travel a given distance varies inversely as the velocity. Suppose the distance is 360 miles: then,

if the velocity is 12 miles per hour, the time will be 30 hours;

that is, if the velocity is doubled, the time is halved. The one varies inversely as the other.

Conversely, if one quantity varies inversely as another, the product of the two quantities is constant.

then the space (S) is a constant quantity.

(233.) One quantity may vary directly as a second, and inversely as a third. Thus, according to the Newtonian law of gravitation, the attraction (G) of any heavenly body varies

directly as the quantity of matter (Q), and inversely as the square of the distance (D).

(234.) Application of the preceding principles.

Ex. 1. Given x+y:x::5:3, } to find the values of x and y.

Since

xy=6,

x+y:x::5: 3.

Substituting this value of y in the second equation, we obtain

Ex. 2. Given x+y: x-y::3: 1, to find the values of x

From the first equation, by Art. 222, we obtain

Substituting this value of x in the second equation, we ob

Ex. 3. Given x+y: x−y :: 64 : 1, to find the values of x

By Art. 223,

By Art. 222,

whence

xy=63,

}

x+y: x-y:: 8:1.

2x: 2y:9:7;

xy::9:7.

9y

and y.

Therefore,

x= 7°

Substituting this value of x in the second equation, we ob

tain

y=7, x=±9.

By division, Art. 220, 3xy×(x—y): x−y :: 60 : 1.

Ex. 7. Given x+√x: x-√x:: 3√x+6:2x, to find the values of x.

Ans. x=9, or 4.

Ex. 8. What number is that to which, if 1, 5, and 13 be sev erally added, the first sum shall be to the second as the second to the third?

Ans. 3.

Ex. 9. What number is that to which, if a, b, and c be severally added, the first sum shall be to the second as the second to the third?

Ex. 10. What two numbers are those whose difference, sum and product are as the numbers 2, 3, and 5 respectively? Ans. 2 and 10.

Ex. 11. What two numbers are those whose difference, sum, and product are as the numbers m, n, and p?

Ex. 12. Find two numbers, the greater of which shall be to the less as their sum to 42, and as their difference to 6.

Ans. 32 and 24.

Ex. 13. Find two numbers, the greater of which shall be to the less as their sum to a, and their difference to b.

Ex. 14. There are two numbers which are in the ratio of 3 to 2, the difference of whose fourth powers is to the sum of their cubes as 26 to 7. Required the numbers.

Ans. 6 and 4. Ex. 15. What two numbers are in the ratio of m to n, the difference of whose fourth powers is to the sum of their cubes

SECTION XIV.

PROGRESSION S.

ARITHMETICAL PROGRESSION.

(235.) An Arithmetical Progression is a series of quantities which increase or decrease by the continued addition or subtraction of the same quantity.

Thus, the numbers

1, 3, 5, 7, 9, 11, &c.,

which are obtained by the addition of 2 to each successive term, form what is called an increasing Arithmetical Progression; and the numbers

20, 17, 14, 11, 8, 5, &c.,

which are obtained by the subtraction of 3 from each successive term, form what is called a decreasing Arithmetical Progression.

(236.) To find the last term of an Arithmetical Progression. If a represent the first term of an arithmetical progression, and d the common difference, the successive terms of an increasing series will be

a, a+d, a+2d, a+3d, a+4d, &c.

The successive terms of a decreasing series will be

a, a-d, a-2d, a-3d, a-4d, &c.

Since the coefficient of d in the second term is 1, in the third term 2, in the fourth term 3, and so on, the nth term of the series will be

which

is n.

a±(n-1)d,

may be called the last term when the number of terms Hence,