straightforward way, from the point of view of construction, of representing the formula graphically, but in practice it will frequently be found that two rectilinear scales side by side are more convenient, as they are more compact and quicker and easier to read, the eye not having to follow the line up from one graduation to the curve, and then along to the other graduation.
Fig. 3 has been constructed from fig. 2 to bring ,1-0 mm. ou (. these points. The scale (z 2 ) is the same, while the position of any graduation of the scale (zi) is obtained by dropping a perpendicular to the scale (z2) from the point on the curve in fig. 2 where the vertical line through any value of Zi cuts it.
The two adjacent scales can, however, be constructed directly without che intermediary of a diagram in cartesians, by introducing the idea of the functional scale, which also figures largely in subsequent applications.
If u is the distance of any graduation of the scale (za) from the zero, n the unit employed,
< CI
6-
-0-5
0-LO-O n r . 3.
gives the graduations of the regular or evenly divided scale (z 2 ), in which for equal intervals between the values of z 2 , the intervals between the graduations on the scale are equal.
For the distance of any graduation on the scale (zi) from the zero we have
Fig; 4.
defining the functional scale (zi), in which the graduations are no longer equally spaced for equal intervals in the value of the variable z\, but the segments cut off are proportional to the function /i, although figured with the corresponding values of Zi.
4. Graphic Representation of a Three-Variable Formula in Car- tesian Coordinates. The equation connecting the three variables Zi, z 2 , z 3 of a formula dealing with three variable quantities may, with our notation, be written, in its most general form
/m = o.
We take one of the variables, z 3 say, and give it in turn different values, starting with the lowest value required, and increasing by equal intervals. For each of these values we can construct a curve, as in 2, traced on the network defined by
Proceeding in this way for a suitable number of values of z 3 , a system of iso- plethic curves or isopleths is obtained. Along each of these curves z 3 has a con- stant value, and we mark this value against the curve. Such a diagram is seen sche- matically in fig. 4.
In order to find the value of z 3 corre- sponding to given values of z\, Zj we take a vertical line through the value of z t and a horizontal straight line through the val- ue of z 2 . We then note on what line of the system (z 3 ) the intersection of these two straight lines falls; if it falls between two lines, interpolation by eye is necessary to judge the interme- diate value.
To put it more generally and concisely for values of z lt z 2 , z 3 which satisfy the given equation, the three corresponding lines of the systems (zi), (z 2 ), (z 3 ) meet in a point.
Hence the term " Intersection Nomogram " used to describe dia- grams of this class, as contrasted with the " Alignment Nomo- gram " which will be dealt with later.
The systems of figured lines which it is necessary to employ in diagrams of this sort will in practice be found to render the reading troublesome in comparison with the reading of a simple graduated scale. The intersection of three lines has to be followed back to the place where their values are marked, and the interpolation by eye between the curves is difficult, while if the number of lines is in- creased to facilitate interpolation, the complication and confusion of the whole diagram are increased.
For these reasons, where the form of the equation renders it possible, it is frequently preferable to employ the methods of repre- sentation which will be described later.
5. Principle of Anamorphosis. In the method of representation of the equation
/m=o
described in 4, we took, corresponding to the variables zi, z 2 , evenly divided scales along ox, oy.
Suppose that instead of this we take the functional scales
Instead of the network with evenly spaced meshes corresponding to the evenly divided scales previously employed, we shall now have
a network with unevenly spaced meshes on which the lines of the system will be altered in shape.
Such a transformation, known as an anamorphosis, is only of advantage when it leads to a better arrangement or simplifica- tion of the diagram. Thus it may be resorted to to space out the isopleths which would otherwise be too close together, or to make the curves which constitute them easier to draw and more con- venient for interpolation. A particular case of frequent practical importance is that in which an anamorphosis can transform the iso- pleths into straight lines. This is best illustrated by an example.
Consider the formula
V 2
connecting the retarding force in percentage weight of a train (R), with the speed in miles per hour (V), and the distance of the stop in feet (D).
Taking z\ = D
(a). Fig. 5 shows the representation on the lines of 4, the sys- tem (R) consisting of the parabolas
arranged on the regular network with jtn=o-25 mm., /i 2 =0-625 mm -
ml/hr " 2P
80,
60
1000* Rf. 5.
150? 2000T
(6). If now instead of the network in (a), we employ the network
- = M1 D
y = M2 V 2 we obtain for the system (R) a system of straight lines
y =
Hi 3'34
radiating from the origin (fig. 6).
60S 1005 "IBOO" 200d
Fir. a.
(f). Writing the formula
log D - 2 log V -f log R - log 3-34 = o and employing the network
- =^logD
y = 2M 2 logV
we obtain (fig. 7) for (R) a system of parallel straight lines ^ + log R - log 3-34 = o.
