respectively, find the equation between x and y.

7. If x2 + y2 ∞ z2, and if when x2 + y2 = a2, z = c, find the relation between xyz.

8. If y ox", and if when x = 1, y = 2, and when x = the value of n.

4, y = 32, find

9. If a vary as by, and if x = 3 when y = 2, show that a-s 10. If x2y + xy2 ∞ (x + y)3, show that x x y.

by-2.

11. If the lengths of two variable lines bear to each other a constant ratio, then the sum of the squares of the two lines varies as the rectangle contained by the two lines.

12. If the sum of two variables vary as the difference of their squares, then the difference of the two quantities is constant.

13. If the sum of the squares of two variables vary as their product, then one of the quantities varies as the other.

14. If xy and y2 a vz, then shall xy o vz.

y, show that x2 + y2 ∞ xy, and that 3+ y3

17. If x vary as y, and y vary as z, show that mx + ny + pz varies as a√yz+b√ xz + c√xy, where m, n, p, a, b, c, are constants.

18. If z vary jointly as x and y, and if when z = 1, x = 2, y = 3, find the value of x, when z = 4, y

1.

between x and y in the form of a proportion.

20. Given that the area of a circle varies as the square of its radius, find the ratio between the areas of two circles whose radii are 2 feet and a foot respectively.

21. If a circular plate of metal of 1 foot diameter and 1 inch in thickness be melted down, and cast into another circular plate of one quarter of an inch in thickness, find the radius of the new plate; having given that the volume of a cylinder varies jointly as the square of the radius of its base, and its height.

22. The areas of similar triangles are in the duplicate ratio of their homologous sides; compare the areas of two equilateral triangles, whose sides are 6 feet and 7 feet respectively.

23. The velocity acquired by a body falling from rest varies as the time of its falling from rest; and the velocity of the body at the end of 5 seconds from rest is 161 feet per second; what was its velocity at the end of the first second? Also find how long it must have been falling to acquire a velocity of one mile per minute.

21. The space fallen through by a body from rest varies as the square of the time of its falling. It is found to be 4 seconds falling from a height of 257.6 feet. Hence find the time a stone would be in falling to the bottom of a well 65 feet deep. Also find the relation between the space and time generally.

25. The volume of globe varies as the cube of its radius. If two cannon balls of 3-inch and 4-inch radius respectively be melted down and cast into a single ball, find its radius. Also if the radius of a nine pounder be 2 inches, find the weight of a ball whose radius is 7 inches.

26. The areas of circular sectors vary as the squares of their radii, and the number of degrees in their angles jointly; hence, if the area of a sector of a circle whose radius is 10 feet, and whose angle is 45°, be 39.27 square feet, find the radius of the circle, the area of a quadrant of which is 3.1416 square feet. Also find the area of a circle whose radius is 6 feet.

27. Given that the speed of a train in miles per hour equals the number of horse-powers in the engine diminished by a quantity which varies as the square of the speed, and that when the engine is of 32 horse-power the speed is 30 miles per hour, find what must be the horse-power of the engine when the speed is 20 miles per hour.

CHAPTER XIV.

ARITHMETIC, GEOMETRIC, AND HARMONIC PROGRESSION.

141. Def.-An "Arithmetic Progression" is a series of terms more than two, which go on increasing or decreasing by a common difference.

Note. The difference may be a number positive or negative, as 4 in the 1st of the above; or it may be any algebraical quantity, positive or negative, as b in the 2nd series, and in the last.

y

Also, it is clear that the difference between two adjacent terms is the same from whatever part of the series the two terms be taken.

142. Def.-The quantity which begins the series is called the "First Term," the difference between two adjacent terms is called the " Common Difference," and the amount of all the terms added together is called the "Sum" of the terms.

143. Def.-An "Arithmeticul Mean" between two given quantities is one whose difference from each of them is the same.

Thus the Arithmetical mean between 10 and 16 is 13; that between ab and a + 5b is a + 2b, &c.

All the terms of an Arithmetic Progression, except the first and last are called "Arithmetic means" between the first and last.

Thus,

5, 7, and 9, are said to be 3 Arithmetic means between 3 and 11.

144. In investigating the properties of these series, the use of general symbols is necessary. We shall first consider a particular case, and then proceed to the more general expression.

Now, it is evident from the construction of this, that

That is, any term of the series is found by adding to 5 a number of twos; the number being one less than the number denoting the order of the term in the series.

Now, we wish to express the fact just stated in Algebraical language; we must not, therefore, confine ourselves to any particular term, as the 5th, 6th, 10th, but we must adopt such a representation of the order of the term as shall be true for any term that can be taken.

Let, then, ʼn stand for the order of the term.

By what was said above, the nth term will consist of 5, together with a number of twos, one less than the number n; this number, then, must be denoted by (n − 1), and the quantity to be added must be (n - 1) 2; and the expression for the nth term

5 + (n − 1) 2.

And this is the formula for the general term of the particular series 5, 7, 9, &c.

This, however, is not a general formula for all series, since 5 and 2 are numbers belonging to the particular series 5, 7, 9.

If, however, we take the series

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

This may serve as the general representative of all series, since a may stand for any 1st term, and d for any common difference.

Now, the nth term of this = a + (n − 1) d.

It is usual to denote the nth term by the letter l, we then have

1 = a + (n − 1) d,

(1.) And this formula suffices to find the value of any given term of any given series, by giving to a, n, d, the particular values required.

145. We may regard (1) as an equation of one unknown quantity in every problem to which it applies, three of the quantities, a, n, d are given; the fourth quantity is then the unknown quantity, and may be found by solving the equation in the usual way.

In fact, there are four varieties of problems which can be solved by this formula.

1o. When l, a, n are given, and d is to be found.

Example of the 1st variety, 7, a, n, given.

The first term of a series is 4, the last is 28, and the number of terms is 7 find the common difference.

:

Example of the 2nd variety, l, a, d, given.

The first and last terms are 40 and 80 respectively, and the common difference is 5: find the number of terms.

Examples of the 3rd variety, l, n, d, given.

The last term of a series of 6 terms is 59, and the common difference is 3 find the first term.

To find d, we must subtract one of the terms from the preceding, it

146. We now proceed to find a general expression for the sum of n terms of an Arithmetic progression.

a + (a + d) + (a + 2d) + (a + 3d) + &c.

If we denote the sum by S, we have

S = a + (a + d) + (a + 2d) + +(a + n − 2d) + (a + n − 1d). Also S = (a + n − 1d)+(a+n−2d)+(a+n − 3d)+ +(a+d)+a; if we write the terms in reverse order.

Adding these two equations together, we have, by adding each term to the one vertically over it,

28

=

(2a + n − · 1d) + (2a + n − 1d) + (2a + n − 1d) + (2a + n − 1d) + &c., to n terms;