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it becomes under the present hypothesis

0.x2-0.x = 0

an equation which can be satisfied by the substitution of any number whatever for x. In fact, since the two lights are supposed to be equal in intensity, and to be placed at the same point, they must illuminate every point in the line A B equally.

The solution O, given by the first system, is one of those solutions, infinite in number, of which we have just spoken.

V. Let a = 0, b not being = c.

Each of the two systems in this case is reduced to 0, which proves, that in this case, there is only one point equally illuminated, viz. the point in which the two lights are placed.

The above discussion affords an example of the precision with which algebra answers to all the circumstances included in the enunciation of a problem.

We shall conclude this subject by solving one or two problems which require the introduction of more than one unknown quantity.

Problem 6.

To find two numbers such that when multiplied by the numbers a and b respectively, the sum of the products may be equal to 2 s, and the product of the two numbers equal to p.

Let x and y be the two numbers sought, the equations of the problem will

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Substituting this value in (2) and reducing, we have

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The problem is, we perceive, susceptible of two direct solutions, for s is manifestly √s2-a2bp; but in order that these solutions may be real we must have s2 or = a2b p.

Let a = b = 1; in this case the values of x and y are reduced to

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Here we perceive, that the two values of y are equal to those of x taken in 2 -p represent the value of x, then p will represent the corresponding value of y, and reciprocally.

an inverse order; that is to say, if s+vs

2

We explain this circumstance by observing, that in this particular case the p and the ques

equations of the problem are reduced to x + y = 2 s, x y = tion then becomes, Required two numbers whose sum is 2 s, and whose product is p, or, in other words, To divide a number 2 s into two parts, such that their product may be equal to p.

Problem 7.

To find four numbers in proportion, the sum of the extremes being 2 s, the sum of the means 2 s', and the sum of the squares of the four terms 4 c 2.

Let a, x y, z, represent the four terms of the proportion; by the conditions of the question, and the fundamental property of proportions, we shall have as the equations of the problem

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These four numbers constitute a proportion, for we have

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Prob. 8. What two numbers are those, whose sum is 20, and their product 36 ?

Ans. 2 and 18.

Prob. 9. To divide the number 60 into two such parts, that their product may be to the sum of their squares, in the ratio of 2 to 5.

Ans. 20 and 40.

Prob. 10. The difference of two numbers is 3, and the difference of their cubes Is 117; what are those numbers?

Ans. 2 and 5.

Prob. 11. A company at a tavern had £8 15s. to pay for their reckoning; but, before the bill was settled, two of them left the room, and then those who remained had 10s. a-piece more to pay than before: how many were there in company? Ans. 7.

Prob. 12. A grazier bought as many sheep as cost him £60, and, after reserving 15 out of the number, he sold the remainder for £54, and gained 2s. a head by them; how many sheep did he buy?

Ans. 75.

Prob. 13. There are two numbers, whose difference is 15, and half their product is equal to the cube of the lesser number; what are those numbers ?

Ans. 3 and 18.

Prob. 14. A person bought cloth for £33 15s. which he sold again at £2. 8s. per piece, and gained by the bargain as much as one piece cost him; required the number of pieces ?

Ans. 15.

Prob. 15. What number is that, which, when divided by the product of its two digits, the quotient is 3; and if 18 be added to it, the digits will be inverted?

Ans. 24.

Prob. 16. What two numbers are those, whose sum multiplied by the greater is equal to 77; and whose difference multiplied by the lesser is equal to 12?

Prob. 17. To find a number such, that if you subtract it from ply the remainder by the number itsef, the product shall be 21.

Ans. 4 and 7.

10, and multiAns. 7 or 3.

Prob. 18. To divide 100 into two such parts, that the sum of their may be 14.

square roots Ans. 64 and 36.

Prob. 19. It is required to divide the number 24 into two such parts, that their product may be equal to 35 times their difference. Ans. 10 and 14.

Prob. 20. The sum of two numbers is 8, and the sum of their cubes is 152; what are the numbers? Ans. 3 and 5.

Prob. 21. The sum of two numbers is 7, and the sum of their 4th powers is 641; what are the numbers? Ans. 2 and 5.

Prob. 22. The sum of two numbers is 6, and the sum of their 5th powers is 1056; what are the numbers ? Ans. 2 and 4.

Prob. 23. Two partners, A and B, gained £140 by trade; A's money was 3 months in trade, and his gain was £60 less than his stock; and в's money, which was £50 more than a's, was in trade 5 months; what was A's stock?

Ans. £100.

Prob. 24. To find two numbers such that the difference of their squares may

be equal to a given number, q2; and when the two numbers are multiplied by the numbers a and b respectively, the difference of the products may be equal to a given number, s2

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Prob. 25. To divide two numbers, a and b, each into two parts, such that the product of one part of a by one part of b may be equal to a given number, p, and the product of the remaining parts of a and b equal to another given number, p'.

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Prop. 26. To find a number such that its square may be to the product of the differences of that number, and two other given numbers, a and b, in the given ratio, p: q.

(a+b)p+√(a-b)2p2+4abpq 2(p-g)

Prob. 27. A wine merchant sold 7 dozen of sherry and and 12 dozen of claret for 50%.; he sold 3 dozen more of sherry for 107. than he sold of claret for 67. Required the price of each.

Ans. Claret, 37.; and sherry, 27. per dozen.

Prob. 28. There is a number consisting of two digits, which, when divided by the sum of its digits, gives a quotient greater by 2 than the first digit; but if the digits be inverted, and the resulting number be divided by a number greater by unity than the sum of the digits, the quotient shall be greater by 2 than the former quotient. What is the number?

Ans. 24.

Prob. 29. A regiment of foot receives orders to send 216 men on garrison duty, each company sending the same number of men; but before the detachment marched, three of the companies were sent on another service, and it was then found that each company that remained would have to send 12 men additional, in order to make up the complement, 216. How many companies were in the regiment, and what number of men did each of the remaining companies send on garrison duty?

Ans. There were 9 companies; and each of the remaining 6 sent 36 men.

S

ON THE NATURE OF EQUATIONS.

171. The valuable improvements recently made in the process for the determination of the roots of equations of all degrees, render it indispensably necessary to present to the notice of the student a concise view of the present state of this interesting department of analytical investigation. The researches of Messrs. Atkinson and Horner on the method of continuous approximation to the roots of equations, and the beautiful theorem of M. Sturm for the complete separation of the real and imaginary roots, have given a fresh impulse to this branch of scientific research, and entirely changed the state of the subject of numerical equations. Indeed, the elegant process of Sturm for discovering the number of real roots, and their initial figures in any numerical equation, combined with the admirable method of continuous approximation as improved by Horner, fully complete the theory and numerical solution of equations of all degrees.

We do not intend to enter at great length into the theory of equations; but it is hoped that the portion of it which we have introduced into the present treatise will be discussed in a simple and perspicuous manner, and be found amply sufficient for most practical purposes.

DEFINITIONS.

1. An equation is an algebraical expression of equality between two quantities.

2. A root of an equation is that number, or quantity, which, when substituted for the unknown quantity in the equation, verifies that equation.

3. A function of a quantity is any expression involving that quantity; thus, a x2+b a x2+b, a x3+cx+d, a are all functions of x; and also ax2-by,

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cx+d'

y2+yx+x2+a2+b+2, are all functions of x and y.

These functions are usually written ƒ (x), and ƒ(x, y).

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when divided by x-a, will leave a remainder, which is the same function of a that the given polynomial is of x.

Let f(x)=x"+p x"−1+q x2-2 +

; and, dividing f(x) by x-a, let

Q denote the quotient thus obtained, and R the remainder which does not involve x; hence, by the nature of division, we have

f(x)= Q(x—a) + R.

Now this equation must be true for every value of x; hence, if x=a, we have f(a) = 0 + R;

for R is altogether independent of x, and therefore the remainder R is the same function of a that the proposed polynomial is of x.

EXAMPLES.

(1.) What is the remainder of x2-6x+7 divided by x-2, without actually performing the operation ?

(2.) What is the remainder of a3—6x2+8x-19, divided by x+3?

(3.) What is the remainder of x1+6x3+7x2+5x—4, divided by x—5? (4.) What is the remainder of 3 +p x2+q x+r, divided by x—a ?

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