Cases in which the value of the unknown quantity is indeterminate.

3. In what case would the denominators of the fractional values of the unknown quantities in example 41 of art. 94 become zero? and what is the corresponding absurdity of the enunciation ?

Ans. When an = bm,

that is, when a : b = m : n ;

and the absurdity is, that the ratio of two num-
bers should not be changed by increasing them
both by the same quantity.

4. In what case would the denominators of the fractional values of the unknown quantities in example 48 of art. 94 become zero? and what is the corresponding absurdity of the enunciation ?

Ans. When a = 0,

and the absurdity is, that the squares of two
equal numbers should differ.

97. Corollary. When the solution of a problem gives for the value of either of its unknown quantities a fraction whose terms are each equal to zero, this value generally indicates that the conditions of the problem are not sufficient to determine this unknown quantity, and that it may have any value whatever. In some cases, however, there are limitations to the change of value of the unknown quantity.

L

EXAMPLES.

1. In what case would both the terms of the fractional value of the unknown quantity in example 25 of art. 94 become zero? and how could this value be a solution?

Cases in which the value of an unknown quantity is indeterminate.

and these equations signify, that the couriers
travel equally fast, and start at the same time;
and, therefore, they remain together, and any
number whatever may be taken as the value of
the unknown quantity.

2. In what case would both the terms of either of the fractional values of the unknown quantity in example 31 of art. 94 become zero? and how could this value be a solution?

and these equations signify, that the bodies move
equally fast, and start from the same place; they,
therefore, remain together, and any number
whatever may be taken as the value of the un-
known quantity. -

But, in this case, all the algebraic values of
the unknown quantity but the first become infi-
nite, as they should, because they are obtained
on the supposition that the second body has
passed round the circle once, twice, &c. oftener
than the first body; which is here impossible.

3. In what case would all the terms of the fractional values of the unknown quantities in example 38 of art. 94 become zero? and how could they, then, satisfy the conditions of the ́ problem?

Ans. When a= b = c;

and these equations signify, that the wines and
the mixture are all of the same value; in what-
ever proportion, therefore, the wines are mixed
together, the mixture must be of the required

Cases in which the value of an unknown quantity is indeterminate.

value. But the values of the unknown quanti

ties are still subject to the limitation that their
sum is n.

4. In what case would the terms of the fractional values of the unknown quantities in example 39 of art. 94 become zero? and how could they, then, satisfy the conditions of the problem?

Ans. When m = n,

and bna = ma;

and these equations signify, that the sum b of the
products of the parts of a multiplied by m = n
is to be equal to the product of a multiplied by
n; and this is, evidently, the case into whatever
parts a is divided.

5. In what cases would all the terms of the fractional values of the unknown quantities in example 41 of art. 94 become zero? and how could they, then, satisfy the conditions of the problem?

for these equations indicate that the two re-
quired numbers are only subject to the condition
that their ratio = a: b.

Secondly. When m = n,

for these equations indicate that the two num-
bers are to be equal; and that they are to remain

equal, when they are increased by c, which would
always be the case.

6. In what case would all the terms of the fractional values of the unknown quantities in example 48 of art. 94 become zero? and how could these values be solutions?

and their equations indicate that the numbers
are to be equal, and that their squares are to be
equal, which is always the case with equal num-
bers.

98. Corollary. When the solution of a problem gives a negative value to either of the unknown. quantities, this value is not generally a true solution of the problem; and if the solution gives no other than negative values for this quantity, the problem is generally impossible.

But, in this case, the negative of the negative value of the unknown quantity is positive; so that the enunciation of the problem can often be corrected by changing it, so that this unknown quantity may be added instead of being subtracted, and the reverse.

EXAMPLES.

1. In what case would the value of the unknown quantity in example 25 of art. 94 be negative? why should it be so? and could the enunciation be corrected for this case?

Ans. When a> b;

that is, when the second courier goes slower
than the one he is pursuing, in which case he
evidently cannot overtaké him; and the enun-
ciation does not, in this case, admit of a legiti-
mate correction.

2. In what case would the values of the unknown quantities in examples 29, 31, 32 be negative? why should this be so? and could the enunciations be corrected for this case?

Cases of negative value of unknown quantity.

Ans. When c> C;

that is, when the first body moves faster than the
second, in which case the second cannot over-
take it.

The enunciation may be corrected for this
case by supposing the bodies to travel in the
opposite direction to that which they are at
present taking, that is, by supposing the first.
body to pursue the second.

Examples 31 and 32 are not, however, impossible in this case; for, from the very nature of their circular motion, the first body is necessarily pursuing the second even in their present direction; the second body must not, however, be considered as a feet or a + c t feet behind the first, but as p a or p (a+c t) feet before

it.

3. In what cases would the values of the unknown quantity in example 33 of art. 94 be negative? why should this be the case? and could the enunciation be corrected for this case?

Ans. First. When C<c,

which is subject to the same remarks as in the * preceding question.

Secondly. When C > c,

and ct> a, or >p+a, or > 2p+a, &c. ; that is, when the first body does not start until the second body has passed it once, or twice, or three times, &c.; and if the bodies were moving in the same straight line, the enunciation would not admit of legitimate correction. As it is, however, the first body is still pursued by the second, and is p+a—ct, 2p+a— ct,&c.