trical science. Types of fundamental importance are eu +e-u
and eu
e- which are so important that the forms (eu +e-u)/2
and (eue-u) /2 have been given special names and tables
of their values have been computed and printed.

The first of

Friction Head in Feet per 1000 Ft. of Pipe

Note:

For open conduits, multiply Hydraulic Radius by 4 to get Equivalent Diameter. Diagram gives nearly same 300 results as Kutters Formula with n =.011.

Diagram of Flow in Clean Cast Iron or Wrought Iron Pipes

Based on the Formula, H, in Feet per 1000 Feet = 0.38

1.86
V
d 1.25

FIG. 108.-A Complicated Example of the Use of Multiple Logarithmic
Paper, Form M6. From Transactions Am. Soc. C. E. Vol. LI.

these is called the hyperbolic cosine of u and the second is called the
hyperbolic sine of u; they are written in the following notation:

cosh u = (eu + e-u)/2, sinh u = (eu - e-u)/2

If x

= a cosh u and y = a sinh u, then squaring and subtracting

FIG. 109.-The Curves of the Hyperbolic Sine and Cosine.

The graphs of y = a cosh x and y = a sinh x were called for in exercises 1, 2, §146. They are shown in Fig. 109. The first of these curves is formed when a chain is suspended between two points of support; it is called the catenary. These two curves

are best drawn by averaging the ordinates of y and the ordinates of y = ex and y =

e-x.

Curves whose equations are of the form y

=ex and y =e ̄*,

on quite a variety of forms for various values of the constants. A good idea of certain important types can be had by a comparison of the curves of Fig. 110 whose equations are:

FIG. 110.-Combinations of Two Exponential Curves. After Steinmetz.

The student should arrange in tabular form the necessary

numerical work for the construction of these curves.

If the second exponent be increased in absolute value, the points of intersection with the y-axis remain the same, but the region of close approach of the curves to each other is moved along the curve = e- to a point much nearer the y-axis. To show this the following curves have been drawn and shown in Fig. 111.

У

FIG. 111.-Combinations of Two Exponential Curves. After Steinmetz.

158.* Damped Vibrations. If a body vibrates in a medium like a gas or liquid, the amplitude of the swings are found to get smaller and smaller, or the motion slowly (or rapidly in some cases) dies out. In the case of a pendulum vibrating in oil, the rate of

decay of the amplitude of the swings is rapid, but the ordinary rate of the decay of such vibrations in air is quite slow. The ratio between the lengths of the successive amplitudes of vibration is called the damping factor or the modulus of decay.

The same fact is noted in case the vibrations are the torsional vibrations of a body suspended by a fine wire or thread. Thus a viscometer, an instrument used for determining the viscosity of lubricating oils, provides means for determining the rate of the decay of the torsional vibration of a disk, or of a circular cylinder

suspended in the oil by a fine wire. The "amplitude of swing" is in this case the angle through which the disk or cylinder turns, measured from its neutral position to the end of each swing.

In all such cases it is found that the logarithms of the successive amplitudes of the swings differ by a certain constant amount or, as it is said, the logarithmic decrement is constant. Therefore the amplitudes must satisfy an equation of the form

A = ae-bt

where A is amplitude and t is time. The actual motion is given by an equation of the form

A study of oscillations of this type will be more fully taken up in