EXAMPLE VI.

To find three numbers, fo that the firft, with half the other two, the fecond with one third of the other two, and the third with one fourth of the other two, may be equal to 34.

Let the numbers be x, y, z, and the equations are;

EXAMPLE VII.

To find a number confifting of three places, whofe digits are in arithmetical proportion; if this number be divided by the fum of its digits, the quotient will be 48; and if from the number be fubtracted 198, the digits will be inverted.

Let the 3 digits be! 1x, y, z

Then the number is 2100x+10y+z

If the digits be in

verted, it is

The digits are in

ar. prop. therefore

By question

By question

From 6 and tranf.

6100x+100+2—198=100%+10y+x

799x=99%+198

Divid. by 99

8x=x+2

From 4

8 and 9

9x-2y-z

12100x+10y+z=48x+48y+48%

52x=38y+47%

52%+10438≈+38+473

The number then is 432, which fucceeds upon trial.

13

It fometimes happens, that all the unknown quantities, when there are more than two, are not in all the equations expreffing the conditions, and therefore the preceding rule cannot be literally followed. The folution, however, will be obtained by fuch fubftitutions as are, used in Ex. 7. and 9. or by fimilar operations, which need not be particularly described.

Corollary to the preceding Rules.

It appears that, in every queftion, there must be as many independent equations as unknown quantities; if there are not, then the question is called indeterminate, because it admit of an infinite number of anmay fwers; fince the equations wanting may be affumed at pleasure. There may be other circumftances, however, to limit the anfwers to one, or a precife number, and which, at the fame time, cannot be directly expreffed by equations. Such are thefe; that the numbers must be integers, squares, cubes, and many others. The folution of

fuch

fuch problems, which are alfo called Diophantine, fhall be confidered afterwards.

SCHOLIUM.

On many occafions, by particular contrivances, the operations by the preceding rules may be much abridged. This, however, must be left to the fkill and practice. of the learner. A few examples are the following.

1. It is often eafy to employ fewer letters than there are unknown quantities, by expreffing fome of them from a fimple relation to others contained in the conditions of the question. Thus, the folution becomes more eafy and elegant. (See Ex. 4. 5.).

2. Sometimes it is convenient to express by letters, not the unknown quantities themselves, but fome other quantities connected with them, as their fum, difference, &c. from which they may be easily derived. (See Ex. 1. of Chap. 5.).

3. In the operation alfo, circumftances will fuggeft a more eafy road than that pointed out by the general rules. Two of the original equations may be added together, or may be fubtracted; fometimes they must be previously multiplied by some quantity, to render such addition or fubtraction effectual, in exterminating one of the unknown quantities, or otherwise promoting the folution. Subftitutions may be made of the values of quantities, in place of quantities themselves, and various other fuch contrivances may be used, which will render the solution much lefs complicated. (See Ex. 3. 7. and 9.).

СНАР.