5. If x and y be unequal and x have to y the duplicate ratio of x+z to y+z, prove that ≈ is a mean proportional between x and y.

6. If a b::p : q, then a+b3 :

a3

:: p2 + q2 : a+b

på

p+q

7. If four quantities are proportionals, and the second is a mean proportional between the third and fourth, the third will be a mean proportional between the first and second.

8. If

(a+b+c+d) (a − b − c + d) = (a − b + c − d) (a + b − c − d),

prove that a, b, c, d are proportionals.

9. Shew that when four quantities of the same kind are proportional, the greatest and least of them together are greater than the other two together.

10. Each of two vessels contains a mixture of wine and water; a mixture consisting of equal measures from the two vessels contains as much wine as water, and another mixture consisting of four measures from the first vessel and one from the second is composed of wine and water in the ratio of 2: 3. Find the proportion of wine and water in each of the vessels.

11. A and B have made a bet, each staking a sum of money proportional to all the money he has. If A wins he will have double what B will have, but if he loses, B will have three times what A will have. All the money between them being £168, determine the circumstances.

12. If the increase in the number of male and female criminals be 1.8 per cent., while the decrease in the number of males alone is 4.6 per cent., and the increase in the number of females is 9.8; compare the number of male and female criminals respectively.

XXVIII. VARIATION.

411. The present chapter consists of a series of propositions connected with the definitions of ratio and proportion stated in a new phraseology, which is convenient for some purposes.

412. One quantity is said to vary directly as another when the two quantities depend upon each other, and in such a manner that if one be changed the other is changed in the same proportion.

Sometimes for shortness we omit the word directly, and say simply that one quantity varies as another.

413. Thus, for example, if the altitude of a triangle be invariable, the area varies as the base; for if the base be increased or diminished, we know from Euclid that the area is increased or diminished in the same proportion. We may express this result by Algebraical symbols thus; let A and a be numbers which represent the areas of two triangles having a common altitude, and let B and b be numbers which represent the bases of these triB angles respectively; then

A

=

α b

And from this we deduce

A α (Art. 392). If there be a third triangle having the same

Bb'

altitude as the two already considered, then the ratio of the number which represents its area to the number which represents its

α

A
B

base will also be equal to . Put=m, then =m and A= mB.

b

Here A may represent the area of any one of a series of triangles which have a common altitude, and B the corresponding base, and m remains constant. Hence the statement that the area varies as the base may also be expressed thus; the area has a constant ratio to the base; by which we mean, in accordance with

Article 392, that the number which represents the area bears a constant ratio to the number which represents the base.

We have made these remarks for the purpose of explaining the notation and language which will be used in the present chapter. When we say that A varies as B, we mean that A represents the numerical value of any one of a certain series of quantities, and B the numerical value of the corresponding quantity in a certain other series, and that A= mB, where m is some number which remains constant for every corresponding pair of quantities.

We will give a formal proof of the equation A= mB deduced from the definition of Art. 412.

414. If A vary as B, then A is equal to B multiplied by some constant quantity.

Let a and b denote one pair of corresponding values of two Α B quantities, and let A and B denote any other pair; then a b B= mB, where m is equal to the

by definition.

Hence A =B=

=

415. The symbol is used to express variation; thus A∞ B stands for A varies as B.

416. One quantity is said to vary inversely as another when the first varies as the reciprocal of the second; see Art. 263.

where m is constant, A is said to vary inversely

417. One quantity is said to vary as two others jointly when, if the former is changed in any manner, the product of the other two is changed in the same proportion.

Or if A = mBC, where m is constant, A is said to vary jointlý as B and C.

418. One quantity is said to vary directly as a second and inversely as a third, when it varies jointly as the second and the reciprocal of the third.

where m is constant, A is said to vary directly

as B and inversely as C.

419. If A ∞ B, and B∞ C, then A ∞ C.

For let A = mB, and B=nC, where m and n are constants; then AmnC; and, as mn is constant, A ∞ C.

420. If A ∞ C, and B∞ C, then A± B ∞ C, and √(AB) ∞ C.

=

For let A=mC, and B=nC, where m and n are constants; then A+B (m + n) C, and A-B= (m-n) C; therefore A + B ∞ C. Also √(AB) = √(mnC3)= C′ √(mn); therefore √(AB) ∞ C.

422. If A∞ B, and C∞ D, then AC BD.

For let A= mB, and C=nD, then AC=mnBD; therefore AC ∞ BD.

423. If A ∞ B, then A" ∞ B".

For let A= mB, then A"=m"B"; therefore A” ∞ B”.

424. If A ∞ B, then AP ∞ BP, where P is any quantity variable or invariable.

For let A= mB, then AP = mBP; therefore AP BP.

425. If AB when C is invariable, and A∞ C when B is invariable, then will A ∞ BC when both B and C are variable.

The variation of A depends upon the variations of the two quantities B and C; let the variations of the latter quantities

take place separately, and when B is changed to b, let ▲ be

A B

changed to a'; then, by supposition, = Now let C be

α

b

changed to c, and in consequence let a' be changed to a; then, by

A very good example of this proposition is furnished in Geometry. It can be proved that the area of a triangle varies as the base when the height is invariable, and that the area varies as the height when the base is invariable. Hence when both the base and the height vary, the area varies as the product of the numbers which express the base and the height.

426. In the same manner if there be any number of quantities B, C, D, &c. each of which varies as another when the rest are constant; when they are all changed, A varies as their product.

EXAMPLES ON VARIATION.

= 1, what

1. Given that y varies as x, and that y = 2 when x= will be the value of y when x= 21

=

2. If a varies as b and a 15 when 63, find the equation between a and b.

3.

Given that z varies jointly as x and y, and that ≈= = 1 when x 1 and y = 1, find the value of z when x = 2 and

y = 2.

=

4. If ≈ varies as mx +y, and if ≈ = 3 when x = 1 and y = 2, and 25 when x = 2 and y = 3, find m.