ρ=Σ+Ψ

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from *Graphic Aids In Engineering Computation* Randolph P. Hoelscher, Joseph Norman Arnold, and Stanley H. Pierce, (McGrew-Hill Book Company, New York, N.Y.) 1952.

**6.1. Utility of Special Slide Rules.** By permitting the motion of
functional scales along parallel lines the same general types of formulas
can be solved as on the straight-scale alignment charts, although the
sliding scales have certain inherent limitations not present in some of
the straight-scale charts. The special slide rule with parallel moving
scales requires no auxiliary index lines and is also more convenient
than the alignment chart in being less bulky. The standard slide rule,
although adaptable to a great variety of problems, is often not as rapid
as a special slide rule developed for a particular problem.

Occasionally, a special slide rule is more accurate than the standard slide rule. As an example, Fig. 6.1 shows a special slide rule for the air-density correction factor represented on a chart in Fig. 3.19. Barometric pressures from 710 to 770 mm are represented on the actual rule by a scale 7 in. long. If an ordinary slide rule having 10-in. scales is used in solving this equation, the entire range from 710 to 770 is contained in a space less than ½ in. in length; thus, at least one more significant figure can be obtained by using the special slide rule. Also, the special slide rule performs the addition of 273 to the temperature, which cannot be done by the ordinary rule.

**6.2. Principles of the Parallel-slide Rule.** The special slide rule and
the ordinary slide rule too, for that matter, operate by adding lengths
proportional to the values of the functions represented on their scales.

If

= a length proportional to a function of**L**_{Q}**Q**= a length proportional to a function of**L**_{R}**R**= a length proportional to a function of**L**_{S}**S**

then a slide-rule scale arrangement such as Fig. 6.2 will solve the equation of lengths.

If * m* is the factor of proportionality, or functional modulus, for all of the
scales, then

where * f(Q)*,

Figure 6.3 is a different arrangement of the same three lengths for which the equation is

or

which is identically the same as Eq. (3) with all signs changed. The geometrical relations for four, five, and six or more functional scales may be obtained in a similar manner.

From this brief explanation of the parallel-slide rule two conclusions may be drawn that are important in the design of special slide rules:

- The same functional modulus,
, must be used on all scales on the same rule.**m** - The lengths may be added in any order;
*i.e.,*the order of the scales on the same rule may be arbitrarily chosen.

The middle-support rule may later appear to be an exception to these conditions, but the middle-support rule is in reality two slide rules on the same frame, and entirely different moduli may be used on the two halves. Because of the interdependence of the two halves of the middle-support rule, the order of scales is not entirely arbitrary.

**6.3. Order and Direction of Scales.** It is possible to devise several
ways for operating the slide-rule scales to add mechanically the lengths

as shown in Figs. 6.2 and 6.3, but it is desirable that the same general method of operation be used on all special slide rules, and the best method of operation for slide rules without a runner is as follows:

Set the value of the second variable opposite the value of the first, and read the result opposite an index on the lower scale (see Fig. 6.4). This corresponds to the method ordinarily used in performing division on the standard slide rule. In the case of four variables no index is used, and the answer is read on the lower scale opposite the value of the third variable. An example to illustrate this will be shown later.

By means of a simple key the directions of scales for this method of operation may be conveniently obtained as follows:

- Write the equation with zero as one member and with the terms of the other member in the order of their desired arrangement on the slide rule.
- Write the functions in a column with alternate plus and minus signs on both sides. The order of functions in the column is the same as their order in the equation, and the index, if required, must be included in its proper place.
- The direction of increasing values of the function is indicated by the arrow placed opposite the sign like the one in front of that function in the equation.

To obtain the order and direction of the scales for the equation

it is first written

then the key is formed

The sign before * Q* in Eq. (7) is

scale increases to the right as shown. The slide rule constructed according to this key is shown in Fig. 6.5. If the position of the index and * R*
scale are interchanged in the key and on the rule, all scales will increase
to the right. An inversion of the slide in the frame is the only thing

necessary to produce this change. The order and direction of scales for
Fig. 6.4 are exactly the same as for Fig. 6.5, and Eq. (7) is different from
the equation of Fig. 6.4 only in the coefficient of * R*. The two slide rules
differ only in the range of numbers chosen and in the scale modulus of
the

If it is desired to solve

the key becomes

and all scales increase to the right as shown in Fig. 6.6. Because we are accustomed to reading scales which increase to the right, it is desirable to arrange the terms in the equation so that most of their scales will increase to the right. However, it is also desirable that the answer scale (the quantity usually solved for) come at the bottom of the rule. Both

of these points should be considered in selecting the scale arrangement for a practical problem.

From the discussion of Fig. 6.6 it is evident that the index of Fig 6.5
is in reality a scale with but a single point, and that point is for ** R = 0**.
Changing the position of the index changes the equation solved on the
rule of Fig. 6.5 by an added constant. If the index is moved 2 units to
the right, the equation solved by the rule becomes

If the scales for * Q*,

or

If the index on the slide rule with logarithmic scales is shifted, the equation solved is changed by a constant multiple, so that instead of the constant 1. in Eq. (11) any other constant may occur. Figure 6.7 shows a slide rule having one index for solving the equation * QR = S* and another
index for

Also of importance is the fact that any coefficients of the logarithmic functions become powers in the product form. For example,

or**6.4. Types of Equations.** The following types of equations may be
applied to slide-rule blanks of the forms illustrated in the examples.

The quantities * f(P)*,

one half of the middle-support rule is then used to solve

and the other half of the rule solves

The two * K* scales are placed on opposite sides of the stationary middle
support, the corresponding values of

by transfer lines, and the numerical values of * K* are omitted from the scales.

An auxiliary function is also used in designing a slip-gage rule. If

then the two equations solved on the slip-gage rule are

and

Figure 6.8 shows how the scales for log * K* and log (

The task of transforming an equation into one of the forms given here requires greater ingenuity and is less subject to rules than any other part of the work. The suggestions given in Art. 5.11 may be helpful.

**6.5. Enumeration of Steps in Design.** A suggested order of attack on
a special slide-rule problem is given herewith, considered only from the
theoretical point of view. Practical suggestions about marking the scales
and pasting them on the blanks are given later.

*Step 1.* Arrange the equation in proper form for preparing a slide rule.
If several of the foregoing forms can be used, choose the one which will
be easiest to operate.

*Step 2.* Arrange the terms in the equation so that the quantity usually
solved for comes last and the signs of the terms alternate as nearly as
possible.

*Step 3.* Choose the limits of the variables and select the modulus so
that the length of the longest scale will not exceed the length of the slide-rule blank. Functional moduli of all scales must be the same except for
a middle-support rule.

*Step 4.* Determine the directions of increasing values of the functions
by arranging the terms in a column in the order in which they are to
appear on the rule, and place arrows with reference to the signs as indicated previously.

Sometimes a rearrangement of the scales or a shift in the position of the index will make it possible to have more of the scales increasing to the right.

*Step 5.* Plot the scales, and paste all of them on the rule; then locate
the index by solving an example. If there is no index, the position of one
of the scales along its axis must be located by solving an example.

*Step 6.* The finished slide rule should have all scales clearly labeled
with the symbol and units of the quantity represented on them. The
date, name of the designer, and a description of the method of operation
should also appear on the rule, and it should not be necessary to turn
the rule end for end to make data on the back readable.

**6.6. Single-slide Rule.** The six steps in the design of a single-slide
rule shown in Fig. 6.9 will be illustrated with the Column Formula,

The limits for the variables are * L*, 1 to 60 ft;

*Step 1.* Write the equation in proper form. First write

Then by taking logarithms

*Step 2.* Arrange terms in the desired order for use on the slide rule.
The form above is satisfactory.

*Step 3.* Choose the modulus so that with the limits given (or chosen
in any problem where not assigned) the length of scale will come within
the length of the slide-rule blank, namely, 14 in.

*Step 4.* Write the key, which indicates that all three functions increase
to the right. The function of * S* is peculiar in that it decreases as

*( * L is in inches in the equation. Changing to feet on the slide rule merely changes
the position of the index. )*

increases, so that the scale for * S* has values increasing to the left. The
logarithmic functions usually encountered increase as the variable
increases, but in this example and in trigonometric scales care must be
used to see that the directions of increasing values of the functions are
obtained from the key and not necessarily the direction of the increasing
value of the variables.

*Step 5.* When a number of special slide rules are to be constructed,
it is a great convenience to have a set of logarithmic scales of various
moduli. The chart in the pocket at the back of the book may be used
to plot logarithmic scales for moduli between 2.5 and 15 in. In this
example the scale modulus for * L* and

For functions other than a constant times the logarithm of the variable, it is usually best to calculate the scale distances for most of the
points. For logarithmic scales of a constant plus the variable and for
some other forms, the logarithmic chart may sometimes be used. The
computations in Table I show how the distances are obtained for plotting
the * S* scale.

Intermediate values on the plotted scale may be obtained with sufficient accuracy by geometrically dividing the distances between the
points computed in the table, or a sector chart may be used. The scale
can be plotted both ways from the position selected for * S* = 9000. The
negative sign indicates that the function (18,000 -

*Step 6.* The desirability of notes giving the equation, units, and
method of operation is obvious. It is also desirable to give any additional information that will make the use of the slide rule more convenient. Ordinarily such notes are placed on the back, but they have
been included in the figure title for these examples. A note is added for
the Column Formula saying that * L/r* should not exceed 120 for primary
members or 200 for secondary members, and an additional scale shows
these two values of

If the ordinary slide rule is used for this equation, at least three distinct operations of the rule are required besides the mental addition of the constant 1. Obviously this rule will save considerable time in computation, and mistakes are not likely to occur.

**6.7. Double-slide Rule.** The construction of this type of rule illustrated by Unwin's equation for the flow of steam in pipes. The equation solved by the special slide rule shown in Fig. 6.10 is

*Step 1.* The equation reduced to proper form is

*Step 2.* The equation as written above has the terms in proper order
for making a slide rule.

Step 3. Upon the basis of scale lengths shown below the functional modulus was chosen as 3 in.

The other scale lengths are less than 12 in., and the computations have
therefore not been shown here. The scale modulus for * P*,

*Step 4.* The key to determine the direction of the scales is

Step 5. All scales except the one for * d* may be plotted directly with
the logarithmic chart at the back of the book. For

Step 6. The material shown in the figure title should be lettered on the back of the rule.

Solving this equation with the ordinary slide rule is tedious and the
results are subject to error because of the complicated function of * d*.
Also, it will be noted that the pipe diameter appearing in the equation
is the actual diameter, while pipes are commonly denoted by their nominal
diameter. The slide rule has been calibrated in terms of nominal diameters, thus avoiding the necessity for a table of actual and nominal sizes
in making computations. The scale for mean pressure,

**6.8. Middle-support Rule.** The equation for transmission-line reactance will be used as an illustration of a middle-support rule shown in
Fig. 6.11. The equation is

*Step 1.* The equation is rewritten in proper form

then the auxiliary function

and

*Step 2.* The terms are rearranged in the following order so that the
top half of the rule solves

and the bottom half solves

*Step 3.* The functional modulus of all scales on the top half of the
rule is 10 in.; on the bottom half it is 7 in.

*Step 4.* The key must be arranged so that the scales for * K* and for
log (

Step 5. The index for 60 cycles is located by solving an example. Since the functional modulus of all scales on the lower half of the rule is 7 in., the index marks for 50 cycles and 25 cycles can be found by using the log chart, with a modulus of 7, setting the value 60 on the chart at the index for 60 cycles, and making marks at 50 and 25, respectively.

Step 6. The instructions for operation of the rule, given in the legend for Fig. 6.11, are ordinarily placed on the back of the rule.

The upper index and the frequency scale would ordinarily be omitted
until after the scales are affixed to the slide-rule blank; then their location
can be accurately determined by solving an example. This slide rule
illustrates a device that is often of value; although the radius of conductor, * R*, appears in the equation, it does not appear on the finished
slide-rule scale. The scale for

This method of marking scales in terms of a function other than the
one appearing in the equation can also be used advantageously in some
aeronautical equations, in equations involving data from a magnetization
curve or steam table (**φ** scale, see Fig. 6.10), and in other cases where the relation between the
variable in the equation and the one it is desired to mark on the scale is
given by a curve Whose equation may or may not be known.

Figure 6.11 shows how this is accomplished. The scale for the function, log * R*, in the equation is first constructed; then the values of the
other variable (wire size in this case) are marked opposite their value in
terms of the variable in the equation. The first scale is cut off when the
scale strips are prepared and pasted on the slide-rule blank.

As a second illustration, the equation which is used for the special slide rule shown in Fig. 6.12 and which occurs in heating and ventilating is as follows:

*Step 1.* Arrange the equation in proper form.

*Step 2.* Write the equation with the terms in the order in which they

are to appear on the rule. This requires the insertion of an auxiliary term, thus making two equations,

which is solved on the top half of the rule, and

which is solved on the bottom half of the rule.

*Step 3.* The choice of modulus, construction of the key, and the
remainder of the solution are left to the reader.

The features of this slide rule that lead to its inclusion in this group of examples are

1. The exponent 1.3, which makes it awkward to solve the equation with the ordinary slide rule, is conveniently taken care of with the special rule.

2. It shows how equations combining both multiplication and subtraction may be solved with a special slide rule.

3. The equation is used only within the limits, * T_{R}* between 60 and 90,

There are many other examples of engineering equations which hold throughout only a limited range of the variables and for which a graphical method of solution can be similarly restricted.

**6.9. Single-slide Slip-gage Rule.** The Francis Weir formula

for the flow of water over a rectangular weir, shown in Fig. 6.13, will be used to illustrate this type of rule. The variables are

= quantity of water discharged, 0.1 to 10 cu ft/secQ

= breadth of weir, 1 to 10 ftB

= head of water over weir, 0.5 to 4 ftH

*Step 1.* Write the equation in proper form. It will be observed that
both subtraction and multiplication appear in this equation as in the

example of Fig. 6.12. An alternate way of preparing a slide rule for
such a problem is illustrated in Fig. 6.13, in which the slip-gage scales
in the groove subtract the constant 1 from * B*/0.2

and the slide rule first solves * B*/0.2

*Step 2.* Arrange the terms as they are to appear on the slide rule,
using the auxiliary variable * K* for

and

Step 3. Choose a modulus to get proper scale lengths. Since ^{5}⁄_{2} log * H*
appears to have the greatest range, determine the modulus for this scale.

This modulus is used for all scales, though a different one could be chosen for the upper half of the rule, including the upper half of the slip-gage.

*Step 4.* The key for these equations is

*Step 5.* Since the indices are fixed at the end of the slide, the position
of the scales along their axes must be determined by solving an example
and gluing the scales so that they satisfy this location of the index.

*Step 6.* Place the information contained in the figure title on the
back of the rule.

The slide rule, Fig. 6.13, is equivalent to a middle-support rule with the stationary middle-support scale in the groove. This form of rule is satisfactory for equations in which the transfer lines run from one logarithmic scale to another. On the middle-support rule where the transfer lines run from a uniform scale to a logarithmic scale, the diagonal transfer lines must sometimes be followed for a long distance, Which would be difficult if the scale were in the groove of a slip-gage rule.

The slip-gage rule is peculiar in that it apparently has no index marks, but from a consideration of the method of operation it is evident that the end of the slide is an index pointing to the scales in the groove.

**6.10. Construction Suggestions.** Suitable slide-rule blanks are not
easily obtained because of the limited demand for them. Very satisfactory blanks can be constructed by gluing together several layers of cardboard and forming the tongues and grooves by varying the widths of
the strips.

All of the special slide rules shown here have been made by gluing paper scales on blanks constructed of Wood. The blanks are 14 in. in length. These rules Work satisfactorily if well-seasoned wood is used and if they are not subjected to extremes of temperature and humidity.

In affixing the scales it is usually better to prepare the scales and then fasten them to the slide-rule blank. The disadvantages of this method are that it is difficult to glue the scales in exactly the right place, as they must be for slide rules without an index, and that the paper on which the scales are constructed changes shape when the glue is applied. On the other hand, if the scales are fastened on the rule first, it is awkward to mark the scales, and if a mistake is made, it is sometimes necessary to glue on new scales and start over again.

It is desirable to apply the glue to the slide-rule blank only, when attaching the scales, and then press the scale in place. A weight of some sort should be placed on the scale until it is dry. Also, the slide and frame should be taken apart for gluing the scales.