with respect to a are p(2-1), and p(√2+1).

Hence, in order that a-2 apv 2+p may be positive or zero as (11) requires, it is necessary that we have

which is the condition that p is positive; therefore, in order that the problem may be possible, it is necessary that

Hence the conditions in (12) are reduced to a single condition,

when this condition is fulfilled the problem has one solution. In the limiting case

a = p(V2-1)

the roots of (8) are equal (412, 2) and the triangle is isosceles since then x = y.

451. We can deduce the two theorems:

1. Of all right-angled triangles which have the same perimeter that triangle is the greatest which is isosceles.

2. Of all right-angled triangles which have the same surface, that which has the smallest perimeter is an isosceles triangle.

452. PROBLEM IV.-Inscribe in a sphere of radius R a cylinder of which the total surface is equivalent to the double of the surface of a circle of radius a.

Let x be the radius of the base of the cylinder and 2y the height. It follows from Fig. 1. that + y2 = R2,

System II is easy to solve, and gives four sets of values for x and y.

453. Discussion.-In order that a set of values of x and y which satisfy system II may be a solution of the given problem, it is necessary and sufficient that these values are real, positive, and less than R. When will these conditions be fulfilled?

Let K2 be a positive value which substituted for x in (4) will satisfy this equation. We take as the value of x the positive square root of K2 and determine the corresponding value of y on putting K for x in the equation

In order that the corresponding value of y may be positive, it is necessary that

When this is true, the values of x and y are less than R, since they satisfy the equation

which is one of the equations of system I which is equivalent to system II, from which these values have been deduced. Therefore, the number of solutions of the problem is the number of real roots, positive and less than a2, which equation (4) can have, considered as an equation of the second degree in x2.

that

or

or

In order that equation (4) may have real roots, it is necessary

(6)

(a2 + 2 R2)2 > 5 a*,

a2+2 R2 a 15,

a2 < R2V 5 + 1.

2

[2411, 1]

When condition (6) is fulfilled, the roots of equation (4) are real; and they are positive, since their sum and product are positive (422, 1, 2).

It remains to determine which of these values are comprised between 0 and a2. To decide this question substitute 0 and a2 for x2 in the first member of equation (4); for x2=0, the first member of

equation (4) reduces to + a', which is positive, and for x2 = a2, it becomes

(7) 5a2a-4 a2R2 + a1=4 a2 (a2 — R2),

which has the same sign as a2 R2.

Two cases can arise:

I. a R<0.

When a R2 is negative, one of the roots of

the equation in 2 lies between 0 and a2 (440), and satisfies the condition (5), that K must be less than a2; the other root is greater than a2 (2440), but will not satisfy the condition (5).

II. a R 0. If a2

the condition of reality (6),

R is positive and at the same time

a2 <R2V5+ 1,

2

is fulfilled, that is to say if a lies between R2 and RV5+1, the

2

roots of the equation in x2 either both lie between 0 and a2 or both are greater than a2 (439). In order that both roots may be less than a2 it is necessary and sufficient that their half-sum

is less than a2, that is, that a2 is greater than tion is fulfilled since we suppose that a2> R2.

R2

2

a2 + 2 R2

5

(8422)

But this condi

454. Résumé. A review of the discussion of this problem leads

to the following results.

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

=2R

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 f.

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?

A

9. A and B together invested $8000 in the same business. allowed his money to remain ten months and received for his investment and gain $4125. B allowed 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 $8,5 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 $134. 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 5:31. 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.