VARIATION.

261. In the investigation of the relation which varying and dependent quantities bear to each other the conclusions are more readily obtained by expressing only two terms in each proportion, than by retaining the four.

But though, in considering the variation of such quantities, two terms only are expressed, it will be necessary for the Learner to keep constantly in mind that four are supposed; and that the operations, by which our conclusions are in this case obtained, are in reality the operations of proportionals.

262. DEF. 1. One quantity is said to vary directly as another, when the two quantities depend wholly upon each other, and in such a manner, that if one be changed, the other is changed in the same proportion*.

Let A and B be mutually dependent upon each other, in such a way, that if A be changed to any other value a, B must be changed to another value b, such that A: a :: B: b; then A is said to vary directly as B.

N. B. When it is simply stated that one quantity 'varies' as another it is always meant that the one 'varies directly' as the other.

Ex. 1. If a man agrees to work for a certain sum per hour, the amount of his wages varies as the number of hours during which he works.

Ex. 2. If the altitude of a triangle be invariable, the area varies as the base. For, if the base be increased, or diminished, the area is increased or diminished in the same proportion, (the area of a triangle being the half of the rectangle under the base and perpendicular. See Euclid, Book 1. Props. 36 and 41.)

The sign placed between two quantities signifies that they vary as each other.

* That Variation is merely an abridgment of Proportion is a point to be carefully borne in mind; for one quantity is said to "vary" as another, not because the two increase and decrease together, but because as one increases or decreases, the other increases or decreases in the same proportion. Thus, if y=√ax, in which x and y are varying quantities and a is invariable, y increases as a increases, and diminishes as a diminishes, but y does not "vary" as x, because, as a increases, y does not increase in the same proportion; for instance, if a be doubled, y is not doubled. ED.

263. DEF. 2.

One quantity is said to vary inversely as another, when the former cannot be changed in any manner, but the reciprocal of the latter is changed in the same proportion.

*

1

4 varies inversely as B, (4), if, when 4 is changed to a,

B

1 1 B

B be changed to b, in such a manner that A: a :: :

A: ab: B.

or

Ex. If the area of a triangle be given, the base varies inversely as the perpendicular altitude.

Let A and a represent the altitudes, B and b the bases, of two or AxB= axb; therefore (Art. 240),

equal triangles; then

1 1

AxB

axb

2

2

A: a b : B :: ::

-

B

264. DEF. 3.

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

Thus A varies as B and C jointly, (ABC), when A cannot be changed to a, but the product BC must be changed to bc, such that Aa BC: bc.

Ex. The area of a triangle varies as its base and perpendicular altitude jointly. Let A, B, P, represent the area, base, and perpendicular altitude of one triangle; a, b, p, those of another; then A=BP, and a = bp; therefore

265. DEF. 4. One quantity is said to vary directly as a second and inversely as a third, when the first cannot be changed in any manner, but the second multiplied by the reciprocal of the third is changed in the same proportion.

B

A varies directly as B, and inversely as C,
inversely as C, (42);

A: a

B b

C

when

: ; A, B, C, and a, b, c, being corresponding values of

C

Ex. The base of a triangle varies as the area directly and the perpendicular altitude inversely. The notation in the last Article

In the following articles, A, B, C, &c. represent corresponding values of any quantities, and a, b, c, &c. any other corresponding values of the same quantities.

266. If one quantity vary as a second, and that second as a third, the first varies as the third.

Let AB, and BC, then shall A-C.

For A: a: B: b, and Bb C c, therefore (Art. 241), A: a :: C: c; that is, A¤C.

In the same manner, if AB, and B∞

then Acc

267. If each of two quantities vary as a third, their sum, or difference, or the square root of their product, will vary as the third.

Let AC and BC, then A+B C; also ABC.

By the supposition,

Aa

Cc :: B: b;

.. Aa

B: b;

alternately A: B :: a b (Art. 243),

by composition or division A± B : B :: a ± b : b ;

alternately A B : a ± b :: B : b :: C : c;

268. If one quantity vary as another, it will also vary as any multiple, or part, of the other.

Let AB, and m be any constant quantity, then because A: a

B

b

:: Bb, A: a :: mB: mb, or A: a :: : (Art. 249); that is,

A∞mB, or ∞

B

m

m m

269. COR. 1. If A vary as B, A is equal to B multiplied by some invariable quantity*.

:

For AamB: mb; altern. A mB :: a: mb; if therefore m be so assumed that A=mB, then in all cases a = mb.

Conversely, if A=mB, and m is invariable, then A∞ B.

270. COR. 2. If we know any corresponding values of A and

B, the constant quantity m may be found.

Let a and b be the two values known, then m =

a

fore known; and, in general, A

= xB.
b

Ex. 1. Let sot, and when t=1 suppose 8=16, then, since 8=mt, 16=m, and 8=16t.

271. If one quantity vary as another, any power or root of the former will vary as the same power or root of the latter.

Let A vary as B, then A: a :: B: b, and by Art. 253, 4" : a" :: Bb"; that is, A" B", where n is whole or fractional.

272. If one quantity vary as another, and each of them be multiplied or divided by any quantity, variable or invariable, the products, or quotients, will vary as each other.

Let A vary as B, and let T be any other quantity. Then, by the supposition, A: a :: B: b;

* By the application of this rule almost every question in Variation is readily solved, since the variation is convertible into an equation, to which the usual rules may be applied. ED.

274. If one quantity vary as two others jointly, either of the latter varies as the first directly and the other inversely.

Let VFT, then by Art. 272, F∞

V
T'

V

or T∞

F

275.

COR.

COR. If the product of two quantities be invariable, those quantities vary inversely as each other.

B∞

Let B×P be constant, or B×P¤1; by division, B¤—.

276. If four quantities be always proportionals, and one or two of them be invariable, we may find how the others vary.

Ex. Let p, q, r, s, be always proportionals, and let p be invariable, then 8 gr. Because psqr (Art. 237), ps-qr; and since p is constant, sqr (Art. 268). If both p and q be invariable, sar.

277. If one quantity vary as a second, and a third as a fourth, the product of the first and third will vary as the product of the second and fourth.

Let A B and CD, then AC¤BD.

278. When the increase or decrease of one quantity depends upon the increase or decrease of two others, and it appears that, if either of these latter be invariable, the first varies as the other; when they both vary, the first varies as their product.