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NOTES ON MATHEMATICS. 3
the sum of 35+15+22 shall be divided by the product of 3X8
which is the same as 72 divided by 24.
Whenever a number or a quantity is placed over a line
and a number or a quantity is placed under the same line it
always indicates that the number or quantity over the line shall
be divided by the number or quantity under the line. Such a
quantity is called a fraction.
The quantity above the line is called the numerator, and
the quantity below the line is called the denominator. A frac-
tion may be either proper or improper. The fraction is proper
when the numerator is smaller than the denominator ; for
instance, f ; but improper if the numerator is larger than the
denominator, for instance, -^ = If.
A fraction can always be considered simply as a problem
in division.
Formulas.
A formula is an algebraic expression for some general
rule, law or principle. Formulas are used in mechanical
books, because they are much more convenient than rules.
Generally speaking, the knowledge of algebra is not required
for the use of formulas, because the numerical values corre-
sponding to the conditions of the problem are inserted for
each letter in the formula except the letter representing the
unknown quantity, which then is obtained by simple arithmet-
ical calculations. It is generally most convenient to begin the
interpretation of formulas from the right-hand side ; for instance,
the formula for the velocity of water in long pipes is
:
v — 8.02J ^ d-
> /. /.
In this formula v represents the velocity of the water in
feet per second.
h represents the "head"* in feet.
d represents the diameter of the pipe in feet.
f represents the friction factor determined by experiments.
/ represents the length of the pipe in feet, and 8.02 is a
constant equal to the square root of twice acceleration due to
gravity.
Assume, for instance, that it is required to find the velocity
of the flow of water in a pipe of 3 inches diameter ( %. foot)
;
the length of the pipe is 1,440 feet, the "head" is 9 feet, and
the friction factor is 0.025.
Inserting in the formula these numerical values, and
for convenience writing the diameter of the pipe in decimals,
we have
:
* In hydraulics the word " head " means the vertical difference between the
level of the water at the receiving end of the pipe and the point of discharge, or
its equivalent in pressure. See Hydraulics, page 443.
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