Let a:bc:d, and e:f=g: h, then ae: bfcg: dh

For = and ď

ae cg

bf dh'

that is, ae: bfcg: dh

The proposition is proved in the same way, whatever is the number of analogies to be compounded.

(376.) The same powers of the terms of an analogy are proportional.'

Let a:bc:d, then an: bn = cn: dn

(377.) The same roots of the terms of an analogy are

proportional.'

Let a:bc:d, then /a: b = x/c: w/ d

For; therefore

d

α

Μα

or

or

2/6

(378.) Any root of any power of the terms of an analogy

For am: ¿m cm : dm; hence (377) an: bn = cn: dn (379.) Let a:bc:d, and let m and n be any two

If p and q be any other two numbers, it may be similarly proved that pa±qb_b

that is, ma ± nb: pa ± qb = mc ±nd: pc ± qd

PROPORTIONAL EQUATIONS.

DEFINITIONS.

(380.) A proportional equation is an equation subsisting between two variable quantities, expressing merely the proportionality of the two members of the equation. An ordinary or absolute equation, such as those formerly treated of, states an equality between the two members."

Thus, if x and y are two variable quantities so related that any two values of x, as x and x", are proportional to the corresponding values of y, namely, and y", then x'x'y':y", which is concisely expressed thus, æ ÷ Y, which expression is called a proportional equation.

As a particular example, let a be the rate of travelling of a stage-coach, or the number of miles travelled in an hour, and y the number travelled in a day; then if x', x", the rates of travelling for any two days, be 7 and 8 miles an hour respectively, the corresponding distances travelled on these two days, or y' and y", are 168 and 192; and 7:8 = 168: 192, or x': x'y': y". The same proportion exists for any other two rates of travelling, and the corresponding number of miles travelled per day, so that the proportion x:x"=y:y" is true generally for any two rates of travelling; and this is expressed by the proportional equation ay, which always implies the proportionality of four quantities, namely, of any two values of x, and the corresponding values of y.

Innumerable examples of proportional equations occur in mathematical and physical science; quantities in the latter science, like those in the former, being represented by letters which express their numerical values, or the number of times that the assumed unit of measure is contained in them.

(381.) If two quantities are so connected that when one of them varies, the other varies in the same proportion, the former is said to vary directly as the other."

Thus, if x and y are so related that any two values of x,

as x and x", and the corresponding values of y, namely, y' and y', are so related that a': x"=y': y', then x and y are said to vary directly, and this relation is expressed by ay.

(382.) If two quantities be so related that when one of them varies, the reciprocal of the other varies in the same proportion, the former is said to vary inversely as the latter." 1 1 y' `y'"

Thus, if x': x" =

: x and y are said to vary in

versely, and this is expressed by a÷

14

y

(383.) If when one quantity varies, the product of other two varies in the same proportion, the former is said to vary as the two latter jointly."

Let y', ', and y", ", be two systems of values of y and z corresponding to x and x", two values of x, then if x' : x'' = y'z' : y"z", x is said to vary as y and z jointly, or x. yz

(384.) If when one quantity varies, the quotient of a second divided by a third also varies in the same proportion, the former is said to vary directly as the second, and inversely as the third."

Let x', y', z', and x", y", z", be two systems of values of x, y, and z, then if x': x"= : x is said to vary di

yy"

(385.) A proportional equation may be converted into an absolute equation, by multiplying one side by some constant quantity.'

Letay be the given equation, then if m be a constant quantity of a proper value, my

For if x, y, and a", y", be any two systems of values of x and y, x':x"=y' : y'' (381), therefore (355) x'y' = x'y'; therefore

If now a"" and y'" be another system of values of x and y,

And the same relation is proved for any other system;

Thus, if, as in a former example (380), x be the rate of travelling per hour of a stage-coach, and y the rate per day,

then x'y' = 1:24; therefore

x

y'

=

Ι

24

=m, and

And this absolute equation is true for any system of values

of x and y. Thus, if x = 10 miles per hour, then 10 =

or y = 24 × 10 240 miles per day.

1

249,

(386.) COR." Hence if any system of values of x and y be known, the constant quantity m can be found."

(387.) If one variable quantity be equal to the product of another by a constant quantity, the former variable quantity varies directly as the latter variable quantity."

Let xmy, then x = y

For amy', and "my"; therefore x': "my': my" =yy"; therefore x = y

(388.) The reciprocals of the members of a proportional equation also form a proportional equation."

(389.) If one quantity vary as another, any multiples

or parts of them also vary as each other."

Let xy, then mx ÷ ny

For x' x'y' : y', therefore mx' : mx"=ny': ny", or

mx ny

And this result is true, whether m and n be integral or fractional.

(390.) COR. 1.-' If one quantity vary as another, it will also vary as any multiple or part of the other."

For when m= 1, x ny

(391.) COR. 2.- Any multiples or parts of a variable quantity vary as each other."

Let x be a variable quantity, then mx ÷ nx

For mx': mx" = na: nx", or mx ÷ nx

When m= 1, æ = n

(392.) If both sides of a proportional equation be multiplied or divided by a constant or a variable quantity, the equation will still exist."

Let xy, then if ≈ be either a constant or a variable

quantity, w≈ ÷ yz, and 2 ÷

For a'

xz

y

: x'y': y", and if z', z", are any two values of z,

When z is constant, ≈′ = 2′′ =z, and the proof is the

same.

(393.) COR. 1.- When one quantity varies directly as another, their ratio is constant."