GENERALIZED PROPORTION, OR VARIATION.

85. Let x be a variable, and let y be connected with x by a constant multiplier m, so that y = mx. When x changes its value, becoming 1, say, y also changes value, becoming y1, so that y1 = mx1.

Dividing one equation by the other,

i.e. any two values of x and the corresponding values of y are in proportion.

Also, if a takes a series of values, 1, x2, x, etc., and the corresponding values of y be y1, Y2, Ys, etc.,

The foregoing relations are indicated by saying that y varies as x, or x varies as y, since the relation is mutual, and they are symbolically expressed by writing yxx, or xxy.

Hence to say that y varies as x, is to say that one is a constant multiple of the other, or that any two values of x and the corresponding values of y are in proportion.

86. If yn •

1

2

y varies as the inverse or reciprocal

of z; or y varies inversely as z.

If y=

=n, y varies directly as x and inversely as z.

x 2

If y = nxz, y varies conjointly as x and z.

Ex. 1. If xxyz, and y varies inversely as z2, and if z = 2 when x= 10, it is required to express x in terms of z.

Ex. 2. The velocity of a body falling from rest varies as the square root of the space passed over, and when the body has fallen 16 feet its velocity is 32 feet. Find the relation between space and velocity.

v = m√s, where v = velocity and s = space.

.. v 32 and s = 16 gives 32 = 4 m.

.. m = = 8, and v=8√s, or v2 = 64 s;

which shows that the velocity varies as the square root of the space fallen through.

Ex. 3. The radius of the earth is r, and the attraction upon a body without varies inversely as the square of the body's distance from the centre. The number of beats made per day varies as the square root of the earth's attraction upon the pendulum. How much will a clock, with a second's pendulum, lose daily if taken to a distance r1 from the earth's centre, r, being greater than r. Let n = the number of seconds in a day = 86400, and let g earth's attraction at the surface.

=

the

Then

... noc.

r

Also, if n

be the number of beats per day made by the pendu

EXERCISE VI. b.

1. The space passed over by a body falling from rest varies as the square of the time, and a body is found to fall 196 feet in 3 seconds. Find the relation between the space and the time.

=

2. If xxy and y = 33 when x 6, find the value of y when = §.

x =

3. y varies inversely as x2, and z varies directly as x2. When x = 2, y + z = 340; and when x = 1, y-z = 1275. For what value of x is y equal to z?

4. zxu- v, u xx, and vxx2. When x = 2, z = 48; and when x = 5, z = 30. For what value of x is z = 0 ?

5. If xy xx2 + y2, and x = 3 when y = 4, find the relation connecting x and y.

6. The area of a rectangle is the product of two adjacent sides; if the area is 24 when the sum of the sides is 10, find the sides of the rectangle.

7. If x + y xxy, then x2 + y2 ∞ xy.

8. If xxy, show that x2 + y2 x xy. ·

9. A watch loses 2 minutes per day. It is set right on March 15th at 1 P.M.; what is the correct time when the watch shows 9 A.M. on April 20th ?

10. The volume of a gas varies directly as its absolute temperature, and inversely as its tension. 1000 cc. of gas at 240° and tension 800 mm. has its temperature raised to 300° and its tension lowered to 600 mm. What volume has the gas then?

11. The attraction at the surface of a planet varies directly as the planet's mass and inversely as the square of its radius. The earth's radius being 3960 miles, and the moon's 1120, and the mass of the earth being 75 times that of the moon, compare the attractions at their surfaces.

12. The length of a pendulum varies inversely as the square of the number of beats it makes per minute, and a pendulum 39.2 in. long beats seconds. When a seconds pendulum loses 30 sec. per day, how much too long is the pendulum?

13. When one body revolves about another by the law of gravitation, the square of the time varies as the cube of the distance. The moon is 240,000 miles from the earth, and makes her circuit in 27 days. In what time would she complete her circuit if she were 10,000 miles distant?

CHAPTER VII.

INDICES AND SURDS.

87. The index law is the result of the convention that when Ρ is a positive integer, a a a...•• to p factors shall be denoted by a". And by this law a2 · a2 = ap+q, P and q both being positive integers.

Now, if algebra is to be consistent with itself we can have only one index law, whatever p may denote, and instead of making a new convention we must interpret in conformity with this index law the cases in which p is not positive and integral.

The interpretation of p zero, or negative and integral, is given in Art. 38. We deal here with p fractional.

(1) If p=q=3, ao · aa = a2 . a1

= a1 =α.

Therefore, a3 is the same as √a, the meaning of which is fully given in Arts. 47 and 48.

Similarly, a. a. a}_a}+} + }

= a.

And a means that a is to be separated into three identically equal factors, and that one of these is to be taken.

and a tells us to separate a into n identically equal factors, and take one of these factors,