CHA P.

III.

PART II.

General Solution of Problems.

N the folutions of the queftions in the

IN

preceding part, the given quantities (being numbers) disappear in the last conclusion, so that no general rules for like cafes can be deduced from them. But, if letters are used to denote the known quantities, as well as the unknown, a general folution may be obtained, because, during the whole courfe of the operation, they retain their original form. Hence also the connection of the quantities will appear in fuch a manner as to discover the neceffary limitations of the data, when there are any, which is effential to the perfect folution of

a problem. From this method, too, it is easy to derive a fyntheticaly demonftration of the folution. derepf ca

When letters, or any other fuch symbols, are employed to exprefs all the quantities, the algebra is fometimes called fpecious or literal.

EXAMPLE WIII.

To find two numbers, of which the fum and difference are given.

Lets be the given fum, and d the given difference; alfo, let x and y be the two numbers fought.

Thus, let the given fum be 100, and the difference 24.

Then x=(s+d=124 — ) 62 aud y . ("—"———)38..

2

In the fame manner may the canon be applied to any other values of s and d. By reversing the steps in the operation, it is ea

s+d

fy to fhew, that if x= and

2

y =

2

the fum of x and y must be s, and their difference d

EXAMPLE IX.

If A and B together can perform a piece of work in the time a, A and C together in the time b, and B and C together in the time c, in what time will each of them perform it alone?

Let A perform the work in the time x, B in y, and C in %; then, as the work is the fame in all cafes, it may be reprefented by unity.

Example in Numbers.

Let a8 days, b=9 days, and c = 10;

34 49

27.

4

appears likewife that a, b, c, must be such, that the product of any two of them muft be lefs than the fum of thefe two multiplied by the third. This is necessary to give pofitive values of x, z, which y, and alone can take place in this queftion. Befides, if x, y, and z, be affumed as any known numbers whatever, and if values of a, b, and c be deduced from fteps 7th, 8th, and 9th, of the preceding operation, it will will have the pro-in

appear

that

b, and a,

perty required in the limitation here men

tioned.

If a, b, and c were fuch, that any of the quantities, x, y, or 2, becarne equal tovo, it implies, that one of the agents did nothing in the work. If the values of any of these quantities be negative, the only fuppofition which could give them any meaning would

be, that fome of the agents, inftead of pro

moting

2