GRANGE, in his Leçons sur le Calcul des Functions, (2d edit. page 223), observes that the result takes

0

place in certain formulæ, when there are cases which they cannot represent; this being, as it were, the means that Analysts employ, to escape from contradictions.

In fact, under the two hypotheses, x=0, and

0

y=0, the two equations are equivalent to one, con

taining two unknown quantities, which admit of an infinite number of solutions, as has been already hinted at (Art. 202).

230. Let it be required, for example, to find the values of x and y in the equations

4x+3y=7;

12x+9y=21.

Here, comparing these equations with equations. (A) and (B), we have a=4, b=3, c=7, a'=12, b'9, and c'=21; then,

Now, by assuming values of a we shall have as many corresponding values of y, which will satisfy the conditions proposed.

above formulæ of the root of y; if x=2, then y=

for every value which we assign to x, we shall have a corresponding value of y; and, consequently, of in this particular case; and as we are not limited in the number of values, which we can assign to x; we may therefore conclude that the number of values that answer the conditions required, are unlimited; but as these values are sometimes confined to whole positive quantities, the number of answers are sometimes limited.

As the consideration of such equations belongs to indeterminate analysis, it is not here necessary to pursue any further these investigations. We can easily see, without the aid of analysis, that the above equations are not independent of one another; for, if the first equation be multiplied by three, it gives the second; and, consequently, the second equation furnishes no new condition.

231. Reciprocally, if the values of x and y pre

0

sent themselves under the form the question is 0'

indeterminate, in fact, we have in this case the three equations

"cb'—bc'=0, ac'—a'c=0, ab'—ba'=0,

of which some one of them is the consequence of the other two; for, from the first we deduce

cb'

b'

which value, substituted in the second, gives the third ab'-ba'=0.

If we take in the first the value of b, and substitute it in the third, we will find the second; if we take in the second a', in order to substitute the value in the third, we will arrive again at the first. This being premised, the first two of these conditions give

we will carry into it at the same time the hypothe sis of the indetermination of the roots, and we shall find, after proper reductions,

ax+by=c.

The two equations reducing themselves to a single one; the question remains therefore indeterminate.

232. Let us examine the case of

c=0, c'=0,

that is to say, that where the known quantities are wanting; then the equations (A) and (B), arc of the form

`ax+by=0, a'x+b'y=0;

it is plain that the first members become nothing, by having x=0, and y=0; a conclusion which would also result from the general values of x and y, since the numerators cb'-bc', and ac'-ca', are nothing in the above hypotheses.

But if we divide the two proposed equations by

y

~, and putting-p, we shall have

Ꮖ

a+bp=0, a'+b'p=0;

α

from the first, we deduce p=- a value which

b'

substituted in the second, gives this equation of condition,

ba-ab=0, or ab'-ba-0,

which is the common denominator of the values of x and y. Therefore, if the numbers c and c', be each nought, the numbers a, b, a', b', are such that this last relation must be satisfied, and we shall have

In order to see then how the two given equations would be modified, when we substitute in a'x+

ab'

b'y=0, for a', its value deduced from ab'

b

ha' 0, this equation becomes the first, so that the two equations make only one. It may be still remarked that, in the actual hypotheses, we can only determine the ratio P, for which we find

so that it is sufficient to take for x and y the same

α α' multiples of the two terms of the fractions or

b

simplified, which explains otherwise the indetermination of these unknown quantities..

It may happen that the two equations,

ax+by=c, a'x+b'y=c',

would be incompatible, or that they express two

contradictory conditions, which should take place, under these relations

a=pa, b' pb, c'=qc;

for then the proposed equations become

ax+by=c, pax+pby=qc:

the second is in opposition to the first, since it expresses an equality between two equal factors multiplied by two unequal factors. The introduction of these hypotheses, upon a', b', c', into the formulæ of roots, gives

_c(p −q)

- c(q-p).

and here this character

0

produces evidently, as announced (Art. 165), a contradiction in the terms of the enunciation.

233. Reciprocally, when the values of x and yare infinite, the two equations are contradictory; we have in order to express this circumstance,

ab-ab0, whence a'

and substituting for a' this value in

a'x-b'y=c',

it becomes, after multiplying by b1

b'(ax+by)=bc',

an equation contradictory to

-- ax + by=c,

=

ab'

since we do not suppose be'=b'e; for then, on account of abab, we should have the consequence acdc, and from these three equations, there would result, as we have seen above,

results which are not those that we supposed.