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equation (4) reduces to + a”, which is positive, and for ac" = a”, it

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I. a”— R* < 0. When a”— R* is negative, one of the roots of the equation in aco lies between 0 and a” (4440), and satisfies the condition (5), that K* must be less than a”; the other root is greater than a” (4440), but will not satisfy the condition (5).

II. a” R* > 0. If a”— R* is positive and at the same time the condition of reality (6),

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is fulfilled, that is to say if a” lies between R* and Roti, the roots of the equation in aco either both lie between 0 and * or both are greater than a” (#439). In order that both roots may be less than a” it is necessary and sufficient that their half-sum **** (3422) is less than a”, that is, that a* is greater than # . But this condition is fulfilled since we suppose that a* > R*. 454. Résumé. A review of the discussion of this problem leads

to the following results.
1. Problem has one solution when 0 <a” < R*.

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Hence it follows that, of all cylinders inscribed in a given sphere, that cylinder has the greatest surface which has the radius

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Geometrical interpretation of the result in 4: The surface expressed by (8) is equivalent to the lateral surface of a cylinder.

PROBLEMs LEADING To EQUATIONs of THE SEcond DEGREE IN Two or MoRE UNKNowN QUANTITIES

1. Find two numbers whose product is 576 and whose quotient is #.

2. The product of two numbers is p and their quotient is q; what are the numbers?

3. Two numbers are in the ratio 11 : 13, and the sum of their squares is 14210. What are the numbers?

4. The product of two numbers multiplied by their sum is 1820, by their difference is 546. What are the numbers?

5. The sum of two numbers and the sum of their squares added gives 686. The difference of the numbers and the difference of their squares added gives 74. What are the numbers?

6. The product of the sum and difference of two numbers is a ; the ratio of their sum to their difference is p : q. What are the numbers?

7. If the figures in a number of two digits are reversed, the new number is 18 less than the given number. The product of the tw numbers is 1008. What are the numbers?

8. A grocer buys $44 worth each of coffee and sugar and receives 90 lbs. more of the latter than of the former. He sells 57 lbs. of sugar and 29 lbs. of coffee at a profit of 20 per cent, receiving $31. How many pounds of sugar and of coffee did he buy?

9. A and B together invested $8000 in the same business. A allowed his money to remain ten months and received for his investment and gain $4125. Ballowed his money to remain eight months and received for his investment and gain $4590. How much money did each invest?

10. 60 lbs. of Java coffee cost $4 less than 60 lbs. of Mocha. A man purchases $81's worth of each kind, and receives 8 lbs. more of Java than of Mocha. Find the cost of each kind of coffee.

11. Find two numbers whose sum is nine times their difference and whose product diminished by the greater number is equal to twelve times the greater number divided by the less.

12. Two workmen were employed at different wages and paid at the end of a certain time; the first received $24 and the second, who had worked 6 days less, received $13}. If the second had worked all the time and the first had lost, 6 days, they would have received the same sum. How many days did each work, and what were the wages of each?

13. A vessel can be filled with water by two pipes; by one of them alone the vessel would be filled 2 hours sooner than by the other; by both pipes together it can be filled in 13 hours. Find the time which each pipe alone would take to fill the vessel.

14. The floor of a room has 273 square feet, one of the walls 189 square feet, and an adjacent wall 117 square feet. How long, broad, and high is the room?

15. The length, breadth, and height of a stone which has rectangular faces have the ratios 5:3:1. The entire surface of the stone is 15.94 square feet. What are the length, breadth, and thickness of the stone?

16. Determine three numbers such that the product of the first and the second is m, the product of the first and third is n, and the product of the second and third is p.

17. Determine four numbers such that the products of any three successive numbers are respectively l, m, n, p.

18. The diagonals of the three rectangular faces of a rectangular parallelopiped, which meet in a vertex of the solid, are respectively a, b, c. What is the area of the three faces?

19. If the first digit of a number containing 6 figures is interchanged with the fourth digit, the second with the fifth, the third with the sixth, a number is formed which, multiplied by the given number, is 122,448,734,694, and which, diminished by the first, gives a remainder which is five times the first number. What is the number?

20. Calculate the sides of a right-angled triangle, given the perimeter 2p of the triangle and that the volumes of the solids generated by revolving the right-angled triangle about the two legs of the triangle is one half of the volume of the sphere whose radius is r. Suggestion.—Let r and y be the legs and z the hypotenuse of the triangle; then the equations will be a + y + z = 2p zy (c-i- y) = 2 r" a”-- y” = z*, etc. 21. Find the four terms of a proportion, given that the sum of the extremes is 21, that of the means 19, and the sum of the squares of the four terms is 442.

22. Find the sides of a right-angled triangle, given the altitude h (hypotenuse being the base), and the difference, c, between the legs.

23. If the sum of any two of three numbers is multiplied successively by the third, the successive products are respectively 810, 680, and 512. What are the numbers?

24. Four quantities are in a proportion. The product of the extremes is a, the sum of the first two terms is b, and the sum of the last two terms is c. What are the four terms?

25. Four quantities are in a proportion. The sum of the first and fourth is a ; the sum of the second and third is b; and the sum of the squares of the four quantities is c. What are the numbers?

CHAPTER XII

GRAPHICAL REPRESENTATION OF THE SOLUTIONS OF SYSTEMS OF SIMULTANEOUS QUADRATIC EQUATIONS

455. Graph of the General Quadratic Function ax” + br-i- c.

This problem is illustrated by the following example: ExAMPLE. –Plot the equation y = z*—4 a -5.

In the table below are arranged the various values of y which correspond respectively to values a = 0, + 1, + 2, . . . . . , — 1, -2, — 3, etc., in the equation y = z*— 4 at – 5.

For y = z*–4a – 5
oc 3/
0 — 5
+ 1 — 8
+ 2 — 9 :
+ 3 — 8 | | | | ||
+4 - — 5 - .
+5 0 ----
+ 6 + 7
etc. etc.
— 1 0
— 2 + 7
s — 3 +16
etc. etc.

FIGURE 1

Note 1.-It is clear that in case the values of a increase beyond those given in the table, each corresponding value of y will be larger than the one preceding; hence the values of a; and y given in the table are sufficient to determine the ultimate directions of the curve. In general, when this is found to be the case, one need not compute more values for the table.

Nore 2. —If the graph of an equation of the second degree in two variables consists of a single branch which extends to a part of the plane at an infinite distance from the origin, it is called a parabola.

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