If any pair of corresponding values of A and B are known, the constant m can be determined.

For instance, if A = 3 when

32.

. . m = 1,

A = B.

DEFINITION. One quantity A is said to vary inversely

as another B, when A varies directly as the reciprocal of B.

Thus if A varies inversely as B, A

=

m

, B

where m is constant.

The following is an illustration of inverse variation: If 6 men do a certain work in 8 hours, 12 men would do the same work in 4 hours, 2 men in 24 hours; and so on. Thus it appears that when the number of men is increased, the time is proportionately decreased; and vice-versâ.

Example 1. The cube root of x varies inversely as the square of y; if x=8 when y=3, find x when y = 1}.

Example 2. The square of the time of a planet's revolution varies as the cube of its distance from the Sun; find the time of Venus' revolution, assuming the distances of the Earth and Venus from the Sun to be 911 and 66 millions of miles respectively.

Let P be the periodic time measured in days, D the distance in millions of miles; we have P2 α D3,

D3.

Hence the time of revolution is nearly 224 days.

33. DEFINITION. One quantity is said to vary jointly as a number of others, when it varies directly as their product.

Thus A varies jointly as B and C, when A = mBC. For instance, the interest on a sum of money varies jointly as the principal, the time, and the rate per cent.

34. DEFINITION. A is said to vary directly as B and inversely as C, when A varies as

B

C'

35. If A varies as B when C is constant, and A varies as C when B is constant, then will A vary as BC when both B and C vary.

The variation of A depends partly on that of B and partly on that of C. Suppose these latter variations to take place separately, each in its turn producing its own effect on A; also let a, b, c be certain simultaneous values of A, B, C.

1. Let C be constant while B changes to b; then A must undergo a partial change and will assume some intermediate value a', where

2. Let B be constant, that is, let it retain its value b, while C changes to c; then A must complete its change and pass from its intermediate value a' to its final value a, where

36. The following are illustrations of the theorem proved in the last article.

The amount of work done by a given number of men varies directly as the number of days they work, and the amount of work done in a given time varies directly as the number of men; therefore when the number of days and the number of men are both variable, the amount of work will vary as the product of the number of men and the number of days.

Again, in Geometry the area of a triangle varies directly as its base when the height is constant, and directly as the height when the base is constant; and when both the height and base are variable, the area varies as the product of the numbers representing the height and the base.

Example. The volume of a right circular cone varies as the square of the radius of the base when the height is constant, and as the height when the base is constant. If the radius of the base is 7 feet and the height 15 feet, the volume is 770 cubic feet; find the height of a cone whose volume is 132 cubic feet and which stands on a base whose radius is 3 feet.

Let h and r denote respectively the height and radius of the base measured in feet; also let V be the volume in cubic feet.

37. The proposition of Art. 35 can easily be extended to the case in which the variation of A depends upon that of more than two variables. Further, the variations may be either direct or inverse. The principle is interesting because of its frequent occurrence in Physical Science. For example, in the theory of gases it is found by experiment that the pressure (p) of a gas varies as the "absolute temperature" (t) when its volume (v) is constant, and that the pressure varies inversely as the volume when the temperature is constant; that is

pat, when v is constant;

From these results we should expect that, when both t and v are variable, we should have the formula

and by actual experiment this is found to be the case.

Example. The duration of a railway journey varies directly as the distance and inversely as the velocity; the velocity varies directly as the square root of the quantity of coal used per mile, and inversely as the number of carriages in the train. In a journey of 25 miles in half an hour with 18 carriages 10 cwt. of coal is required; how much coal will be consumed in a journey of 21 miles in 28 minutes with 16 carriages?

Substituting now the values of t, c, d given in the second part of the question, we have

Hence the quantity of coal is

5 cwt,

q=125

1. If a varies as y, and x=8 when y=15, find x when y=10.

2. If P varies inversely as Q, and P=7 when Q=3, find P when Q=21.

3. If the square of x varies as the cube of y, and x=3 when y=4, find the value of y when x=

1

√3

3

4. A varies as B and C jointly; if A=2 when B=

=

10 and C= " 27

find when A=54 and B=3.

5. If A varies as C, and B varies as C, then A±B and √√AB will

each vary as C.

C

6. If A varies as BC, then B varies inversely as A

7. P varies directly as Q and inversely as R; also P

Q=23; and R =

9

14

: find when P=√√48 and R=√/75.

8. If x varies as y, prove that x2+y2 varies as x2 - y2.

9. If y varies as the sum of two quantities, of which one varies directly as a and the other inversely as x; and if y=6 when x=4, and y=33 when x=3; find the equation between x and y.

10. If y is equal to the sum of two quantities one of which varies as a directly, and the other as x2 inversely; and if y=19 when x=2, or find y in terms of x.

3;

11. If A varies directly as the square root of B and inversely as the cube of C, and if A=3 when B=256 and C=2, find B when A=24

1

1

12. Given that x+y varies as z+ and that x-y varies as zfind the relation between x and z, provided that z=2 when x= -3 and y=1.

13. If A varies as B and C jointly, while B varies as D2, and C varies inversely as A, shew that A varies as D.

14. If y varies as the sum of three quantities of which the first is constant, the second varies as x, and the third as x2; and if y=0 when x=1,y=1 when x= =2, and y=4 when x=3; find y when x=7.

15. When a body falls from rest its distance from the starting point varies as the square of the time it has been falling: if a body falls through 402 feet in 5 seconds, how far does it fall in 10 seconds? Also how far does it fall in the 10th second?