СНАР.

III.

PART II.

General Solution of Problems.

N the folutions of the queflions in the

IN

preceding part, the given quantities (being numbers) disappear in the last conclufion, fo 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, becaufe, during the whole course of the operation, they retain their original form. Hence also the connection of the quantities will appear in fuch a manner as to difcover the neceffary limitations of the data, when there are any, which is effential to the perfect folution of

a

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a problem. From this method, too, it is eafy to derive a fynthetical demonftration of the folution. J

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

EXAMPLE VIII.

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 íoo, and the difference 24.

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 y

2

s-d

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 z; then, as the work is the fame in all cases, it may be represented by unity.

Example in Numbers.

Let a 8 days, b=9 days, and c = 10;

23

then x=143, y=17, and x=23

49

appears likewife that a, b, c, must be such, that the product of any two of them muft be lefs than the fum of these two multi

plied by the third. This is neceffary to give pofitive values of x, y, and %2 which 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 steps 7th, 8th, and 9th, of the preceding operation, it will appear that b, and a, will have the property required in the limitation here mentioned.

If a, b, and c were fuch, that any of the quantities, x, y, or 2, became equal tò z, 0, it implies, that one of the agents did nothing in the work. If the values of any of thefe quantities be negative, the only fuppofition which could give them any meaning would be, that fome of the agents, inftead of pro