11. If one quantity vary as the sum of two others when their difference is constant, and also vary as their difference when their sum is constant, shew that when these two quantities vary independently, the first quantity will vary as the difference of their squares.

12. Given that the volume of a sphere varies as the cube of its radius, prove that the volume of a sphere whose radius is 6 inches is equal to the sum of the volumes of three spheres whose radii are 3, 4, 5 inches.

13. 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 one inch thick. Find its diameter, having given that the area of a circle varies as the square of its diameter.

14. There are two globes of gold whose radii are r and r'; they are melted and formed into a single globe, Find its radius.

15. If x, y, z be variable quantities such that y+z-x is constant, and that (x + y − z) (x+z-y) varies as yz, prove that x + y + z varies as yz.

16. A point moves with a speed which is different in different miles, but invariable in the same mile, and its speed in any mile varies inversely as the number of miles travelled before it commences this mile. If the second mile be described in 2 hours, find the time occupied in describing the nth mile.

17. Suppose that y varies as a quantity which is the sum of three quantities, the first of which is constant, the second varies as x, and the third as x2. And suppose that when x a, y = 0, when x = 2a, y = a, and when x = 3a, y = 4a. Shew that when

18. Assuming that the quantity of work done varies as the cube root of the number of agents when the time is the same, and

varies as the square root of the time when the number of agents is the same; find how long 3 men would take to do one-fifth of the work which 24 men can do in 25 hours. (See Art. 426.)

XXIX. SCALES OF NOTATION.

427. The student will of course have learned from Arithmetic that in the ordinary method of expressing integer numbers by figures, the number represented by each particular figure is always some multiple of some power of ten. Thus in 347 the 3 represents 3 hundreds, that is, 3 times 102; the 4 represents 4 tens, that is, 4 times 10'; and the 7 which represents 7 units, may be said to represent 7 times 10o.

This mode of representing numbers is called the common scale of notation, and 10 is said to be the base or radix of the common scale.

428. We shall now prove that any positive integer greater than unity may be used instead of 10 for the radix, and shall shew how to express a number in any proposed scale. We shall then add some miscellaneous propositions connected with this subject.

The figures by means of which a number is expressed are called digits.

When we speak in future of any radix we shall always mean that this radix is some positive integer greater than unity.

429. To shew that any positive integer may be expressed in terms of any radix.

Let N denote the the highest power of by r", and let p1 be the

number, r the radix. Suppose that r" is
which is not greater than N; divide N
quotient and N, the remainder; thus
N = p2r" + N1.

Here, by supposition, p, is less than r; also N, is less than r". Next divide N1 by r”-1, and let P1-, be the quotient and N, the remainder; thus

1

2

Proceed in this way until the remainder is less than r; thus we find N expressed in the manner indicated by the equation

430. To express a given integer number in any proposed scale.

By a given integer number we mean a number expressed in words or else expressed by digits in some assigned scale. If no scale is mentioned, we understand the common scale to be intended.

Let N be the given number, r the radix of the scale in which it is to be expressed. Suppose Po P... P1 to be the required digits by which N is expressed in the new scale, beginning with that on the right hand; then

we have now to find the value of each digit.

Divide N by r, and let Q, denote the quotient; then it is obvious that

and that the remainder is p.. Hence p, is found by this rule; divide the given number by the proposed radix, and the remainder is the first of the required digits.

Again, divide Q, by r, and let Q, denote the quotient; then it is obvious that

and that the remainder is p. Hence the second of the required digits is ascertained.

By proceeding in this way we shall determine in succession all the required digits.

431.

For example, transform 43751 into the scale of which 6 is the radix. The division may be performed and the remainders noted thus:

Thus,

6) 4 37 51

6) 7 29 1......5

6) 1 2 15......1

6) 202......3

6) 3 3......4

5......3

43751 = 5.65 +3.6*+ 4.63 + 3.62 + 1.6 + 5,

so that the number is expressed in the new scale thus, 534315.

432. Again, transform 43751 into the scale of which 12 is the radix.

In expressing the number in the new scale we shall require a single symbol for eleven; let it be e; then the number is expressed in the new scale thus, 2139e.

We cannot of course use 11 to express eleven in the new scale, because 11 now represents 1.12+1, that is, thirteen.

433. We will now consider an example in which a number is given, not in the common scale.

A number is denoted by t347e in the scale of which twelve is the radix, it is required to express it in the scale of which eleven is the radix.

Here t stands for ten, and e for eleven.

e) t 3 4 7 e

e 2 7 3......2

The process of division by eleven is performed thus. First e is not contained in t, for eleven is not contained in ten, so we ask how often is e contained in t3; here t stands for ten times twelve, that is one hundred and twenty, so that the question is, how often is eleven contained in one hundred and twenty-three? the answer is eleven times, with two over. Next we ask how often is e contained in 24; that is, how often is eleven contained in twenty-eight; the answer is twice, with six over. Then how often is e contained in 67; that is, how often is eleven contained in seventy-nine; the answer is seven times, with two over. Lastly, how often is e contained in 2e; that is, how often is eleven contained in thirty-five? the answer is three times, with

two over.

Hence 2 is the first of the required digits.

The remainder of the process we will indicate; the student should carefully work it for himself, and then compare his result with that here given.

e) e 273

e) 10 2 t......1

e) 1 1 4......2

e) 1 2......6

1......3