and when A 10, B = 2, find the value of

B when A

B

5. If A ∞ and when A6, B=4, and C = 3, find

C'

the value of A when B = 5 and C=7.

6. If the square of X vary as the cube of Y, and X=3 when Y=4, find the equation between X and Y.

7. If the square of X vary inversely as the cube of Y, and X=2 when Y=3, find the equation between X

and Y.

8. If Z vary as I directly and Y inversely, and if when Z=2, X=3, and Y=4, find the value of Z when X= 15 and Y= 8.

9. If A ∞ B+c where c is constant, and if A 2 when B 1, and if A = 5 when B = 2, find A when B = 3.

=

10. The velocity acquired by a stone falling from rest varies as the time of falling; and the distance fallen varies as the square of the time. If it be found that in 3 seconds a stone has fallen 145 feet, and acquired a velocity of 96 feet per second, find the velocity and distance at the end of 5 seconds.

11. If a heavier weight draw up a lighter one by means of a string passing over a fixed wheel, the space described in a given time will vary directly as the difference between the weights, and inversely as their sum. If 9 ounces draw 7 ounces through 8 feet in 2 seconds, how high will 12 ounces draw 9 ounces in the same time?

12. The

space will vary also as the square of the time. Find the space in Example 11, if the time in the latter case be 3 seconds.

13. Equal volumes of iron and copper are found to weigh

77 and 89 ounces respectively. Find the weight of

10 feet of round copper rod when 9 inches of iron rod of the same diameter weigh 31 ounces.

14. The square of the time of a planet's revolution varies as the cube of its distance from the sun. The distances of the Earth and Mercury from the sun being 91 and 35 millions of miles, find in years the time of Mercury's revolution.

15. A spherical iron shell 1 foot in diameter weighs of what it would weigh if solid. Find the thickness

of the metal, it being known that the volume of a sphere varies as the cube of its diameter.

16. The volume of a sphere varies as the cube of its diameter. Compare the volume of a sphere 6 inches in diameter with the sum of the volumes of three spheres whose diameters are 3, 4, 5 inches respectively.

17. Two circular gold plates, each an inch thick, the diameters of which are 6 inches and 8 inches respectively, are melted and formed into a single circular plate 1 inch thick. Find its diameter, having given that the area of a circle varies as the square of its diameter. 18. The volume of a pyramid varies jointly as the area of its base and its altitude. A pyramid, the base of which is 9 feet square, and the height of which is 10 feet, is found to contain 10 cubic yards. What must be the height of a pyramid upon a base 3 feet square, in order that it may contain 2 cubic yards?

CHAPTER XXI.

SERIES.

365. A succession of numbers which increase or decrease according to some fixed law is called a series; and the successive numbers are called the terms of the series.

1 Thus, by executing the indicated division of the series 1 + x " 1 -x + x2 + x3 +......... is obtained, a series that has an unlimited number of terms.

366. A series that is continued indefinitely is called an infinite series; and a series that comes to an end at some particular term is called a finite series.

367. When x is <1, the more terms we take of the infinite series 1+x+x2+x-3 +......., obtained by dividing 1 by 1−x, the more nearly does their sum approach to the value 1

of

+1 +27 +..........., a sum which cannot become equal to however great the number of terms taken, but which may be made to differ from by as little as we please by increasing indefinitely the number of

terms.

368. But when x is > 1, the more terms we take of the series 1+x+x2+x3+..... the more does the sum of the series diverge from the value of

1

1 Xx

1 -X 1 3

3 +9 +27 +....., a sum which diverges more and more from

the more terms we take, and which may be made to increase indefinitely by increasing indefinitely the number of terms taken.

369. A series whose sum as the number of its terms is indefinitely increased approaches some fixed finite value as a limit is called a converging series; and a series whose sum increases indefinitely as the number of its terms is increased, is called a diverging series.

370. When x= 1, the division of 1 by 1 x, that is, of 1 by 0, has no meaning, according to the definition of division; and any attempt to divide by a divisor that is equal to zero leads to absurd results.

Thus,

by transposing,

8+4; 8-84-4;

or, dividing by 4 - 4, 2=1; a manifest absurdity.

371. When x=1 very nearly, then the value of

1

1-x

will be very great, and the sum of the series 1+x+x2+ 23+..... will become greater and greater the more terms we take. Hence, by making the denominator 1x approach indefinitely to zero, the value of the fraction made to increase at pleasure.

1

1-x

may be

372. If the symbol be used to denote a quantity that is less than any assignable quantity, and that may be considered to decrease without limit, not, however, becoming 0, and the symbol o be used to denote a quantity that is greater than any assignable quantity, and that may be considered to increase without limit, then

ator and denominator will each become 0, and the fraction

374. If, however, x in this fraction approach to 1 as its limit, then the denominator 1-x, inasmuch as it has some value, even though less than any assignable value, may be used as a divisor, and the result is 1+x+x2+x2+x*. Hence, it is evident that though both terms of the fraction become smaller and smaller as 1-x approaches to 0, still the numerator becomes more and more nearly five times the denominator.

It may be remarked that when the symbol is obtained for the value of the unknown quantity in a problem, the meaning is that the problem has no definite solution, but that its conditions are satisfied if any value whatever be taken for the required quantity; and if the symbol %, in which a denotes any assigned value, be obtained for the value of the unknown quantity, the meaning is that the conditions of the problem are impossible.

375. The number of different series is unlimited, but the only kinds of series that can be considered in a work of this character are Arithmetical, Geometrical, and Harmonical Series.

ARITHMETICAL SERIES.

376. A series in which the difference between any two adjacent terms is equal to the difference between any other two adjacent terms, is called an Arithmetical Series or an Arithmetical Progression.

377. The general representative of such a series will be a, a+d, a +2d, a +3d.....,

in which a is the first term and d the common difference;