The Project Gutenberg eBook #36640: Lectures on Elementary Mathematics

M ₊ N

= A,

x² (a − x)²

or

^M + ^N − A = 0.

x² (a − x)²

We will now consider the curve having the equation

^M + ^N − A = y

x² (a − x)²

in which it will be seen at once that by giving to x a very small value, positive or negative, the term ^M , while continuing posi-

_x2

tive, will grow very large, because a fraction increases in propor-

tion as its denominator decreases, and it will be infinite when

M

x = 0. Further, if x be made to increase, the expression

constantly diminish; but the other expression ^N

(a − x)²

_x2 will

, which

was ^N

_a2

when x = 0, will constantly increase until it becomes

very large or infinite when x has a value very near to or equal

to a.

Accordingly, if, by giving to x values from zero to a, the sum of these two expressions can be made to become less than the given quantity A, then the value of y, which at first was very large and positive, will become negative, and afterwards again become very large and positive. Consequently, the curve will cut the axis twice between the two lights, and the problem will have two solutions. These two solutions will be reduced to a single solution if the smallest value of

M ₊ N

x² (a − x)²

is exactly equal to A, and they will become imaginary if that value is greater than A, because then the value of y will always be positive from x = 0 to x = a. Whence it is plain that if one

of the conditions of the problem be that the required point shall fall between the two lights it is possible that the problem has no solution. But if the point be allowed to fall on the prolongation of the line joining the two lights, we shall see that the problem is always resolvable in two ways. In fact, supposing x negative, it is

plain that the term ^M

_x2

will always remain positive and from be-

ing very large when x is near to zero, it will commence and keep

decreasing as x increases until it grows very small or becomes

zero when x is very great or infinite. The other term ^N ,

(a − x)²

which at first was equal to ^N , also goes on diminishing until

_a2

it becomes zero when x is negative infinity. It will be the same

if x is positive and greater than a; for when x = a, the expres-

sion ^N

(a − x)²

will be infinitely great; afterwards it will keep on

decreasing until it becomes zero when x is infinite, while the

other expression ^M

_x2

will first be equal to ^M

_a2

and will also go on

diminishing towards zero as x increases.

Hence, whatever be the value of the quantity A, it is plain that the values of y will necessarily pass from positive to nega- tive, both for x negative and for x positive and greater than a. Accordingly, there will be a negative value of x and a positive value of x greater than a which will resolve the problem in all cases. These values may be found by the general method by successively causing the values of x which give values of y with contrary signs, to approach nearer and nearer to each other.

With regard to the values of x which are less than a we have seen that the reality of these values depends on the smallest

value of the quantity