2. Graphic Representation of a Two-Variable Formula in Cartesian Coordinates

With the functional notation explained above, the most general expression for an equation connecting two variables z 1, z 2, is, fit =o.

In the case of a practical formula, supposing z 2 to be the quantity which usually has to be determined for values of z 1, we as a rule have the equation in the explicit form, z2 =f1.

Taking the rectangular axes Ox, Oy (fig. r) we construct the curve C, the " graph " of z2 =A the abscissa x and ordinate y of any point on this curve representing corresponding values of z i, z 2 respectively.

Suitable scales are selected for z i and z 2, according to the size of the diagram and the range of values of z i, z 2 required. Then, denoting by µl, µ2 the units of the scales (z i), (z2), x =?i z1 Y = define the graduations of the scales ,(z 1), (z 2) along Ox, Oy.

If any two corresponding values of z 1, z 2 are taken and parallels to Ox, Oy drawn through the appropriate graduations on their respective scales, the intersection of these two straight lines gives a point on the curve.

Proceeding in this way with different corresponding values of z1 i 1 2, the necessary number of points on the curve to enable it to be constructed with sufficient accuracy are obtained.

Having constructed the curve in this manner, the value of z2 corresponding to any value of z 1 is obtained by following the parallel to Oy through the given value of z 1 on the scale (z i) till it cuts the curve, and then following the parallel to Ox through this point till it cuts the scale (z 2) at a certain graduation. This graduation gives the value of 1 2 required.

In order to save the trouble of having to draw the parallels on the diagram each time a reading is required, we construct a sufficient number once for all through the graduations of the scales (z i), (z2) so that the eye can follow them and, if necessary, interpolate between them to read the corresponding values.

Looking at the matter in a slightly different way, x = ?1z1 y =1.1212 may be considered as defining two systems (z i), (z2) of parallel straight lines at right-angles to each other, forming a rectangular network, the vertical and horizontal " meshes " of which are " figured " to correspond with the graduations of the scales through which they are drawn.

For any two values of z 1, 1 2 which satisfy z2 =fi we will then have two corresponding straight lines in this network which will intersect on the curve 12 =./.1.

In practice the familiar " squared paper," already prepared with rulings at intervals of a millimetre or a tenth of an inch, is largely employed for work of this sort. Fig. 2 shows such a diagram constructed for the electrical formula d / giving the current (C) in amperes which will fuse a wire of diameter d mm., K being a constant depending on the metal of the wire. In this case the diagram has been constructed for lead wire (K = ro8) of thickness up to i mm., and z 1 =d, Ai =50 mm.

22 = C, 142 = 5 mm.

3. Graphic Representation of a Two-Variable Formula by Means of Two Adjacent Scales. - The method described in §2 is the most amps 5 / c 1 d -'UoelT (C) 1 0 mm.

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 out these points. The scale (z2) is the same, amps 10, while the position of any graduation of the scale (z l) is obtained by dropping a perpendicular to the scale (z2) from the point on the curve in r fig. 2 where the vertical line through any value of z 1 cuts it.

The two adjacent scales can, however, be constructed directly without the 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 o 5 scale (z 2) from the zero, ,u the unit employed, u = µZ2 gives the graduations of the regular or evenly divided scale (z2), in which for equal intervals between the values of z 2, the intervals between the graduations on the scale are equal.

_ oo For the distance of any graduation on the o scale (z1) from the zero we have u = µf1 defining the functional scale (z1), in which the graduations are no longer equally spaced for equal intervals in the value of the variable zl, but the segments cut off are proportional to the function although figured with the corresponding values of z1.

4. Graphic Representation of a Three-Variable Formula in Cartesian Coordinates

The equation connecting the three variables 21, z2, z 3 of a formula dealing with three variable quantities may, with our notation, be written, in its most general form f123= 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 x =µ1z1 Y =µ2z2 Proceeding in this way for a suitable number of values of 2 3, a system of isoplethic curves or isopleths is obtained. Along each of these curves z 3 has a constant value, and we mark this value against 3 the curve. Such a diagram is seen sche matically in fig. 4.2 In order to find the value of 2 3 corresponding to given values of zl, z 2 we take a vertical line through the value of 2 1 and a horizontal straight line through the value of z2. We then note on what line of the system (z3) the intersection of these two straight lines falls; if it falls between two lines, interpolation by eye is necessary to judge the intermediate value.

To put it more generally and concisely for values of z l, 2 2, z5 which satisfy the given equation, the three corresponding lines of the systems (zl), (22), (z3) meet in a point. Hence the term " Intersection Nomogram " used to describe diagrams of this class, as contrasted with the " Alignment Nomogram 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 increased 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 representation which will be described later.

5. Principle of Anamorphosis

In the method of representation of the equation 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 simplification 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 convenient for interpolation. A particular case of frequent practical importance is that in which an anamorphosis can transform the isopleths into straight lines. This is best illustrated by an example. Consider the formula R=3.341-5 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 (a). Fig. 5 shows the representation on the lines of §4, the system (R) consisting of the parabolas / / R C / .L 2/ 2 334f2l arranged on the regular network x=µ1D y =µ2V with µ 1. mm., /22 = 0.625 mm.

If now instead of the network in (a), we employ the network x =21D y= µ2V2 we obtain for the system (R) a system of straight lines /2 3.34 Cµ l ? radiating from the origin (fig. 6).

600 1500 20 Fig: 6.

(c). Writing the formula log D - log V + log R - log 3'34 = 0 and employing the network x =µ1 log D y = 2112 log V we obtain (fig. 7) for (R) a system of parallel straight lines x_ _ y - - { - logR - l og3.34=o500 1000' 5.

mm.

(d) Fig. 3.

f123=0 described in 4, we took, corresponding to the variables z1, 22, evenly divided scales along ox, oy.

Suppose that instead of this we take the functional scales x= /-llfl y = l-12f2.

Instead of the network with evenly spaced meshes corresponding to the evenly divided scales previously employed, we shall now have = D = V 2 3 = R ' A logarithmic anamorphosis as illustrated in (c) is so frequently resorted to in practice that paper already ruled with a logarithmic network can be obtained commercially and is largely employed.

6. Graphic Representation of a Three-Variable Formula in Parallel Coordinates. - The preceding sections have dealt with Intersection Nomograms in which the answer is read from the intersection of lines in a point. For certain types of formulae, however, a representation is possible in which the three variables are arranged along three scales, and the answer is read by the alignment of points on these scales. Such an arrangement, an " Alignment Nomogram," is possible only when a diagram for the formula can be constructed, in cartesian coordinates, which consist of three systems of straight lines.

Defining the three systems of straight lines in cartesian coordinates by xfi+ygl+hl = o x { f ' 2+ yg 2-i 7h_2 = o x, J 3 +yg3 +/?3 corresponding to the three variables z 1, z 2, z 3, we arrive at an equation for the formula which is most conveniently expressed in determinant notation fi gl hl f2 g2 h2 - o (i).

f3 g3 h3 For further investigation it is necessary to introduce the idea of Parallel Coordinates referred to two parallel axes, so that a point is represented by an equation of the first degree.

These coordinates are defined as follows: - If a straight line MN (fig. 8) cuts two parallel axes Au, B y (A and B being the origins of the axes) in M and N, the coordinates of the straight line are N =AM, v=BN.

Any equation of the first degree au+bv+c = o will represent a point, and to determine this e point it is sufficient to know two solutions of the equation, and take the intersection of the straight lines resulting from these two solutions.

a _ b Along the axes Au, B y (fig. 9) take AQ = - a, BR= - The intersection of the straight lines AR, BQ in P will then give the point required, the point au+bv+c =0 This correspondence of points to straight lines and vice versa, according as to whether cartesian or parallel coordinates are employed for the geometrical interpretation, e. is an example of the Principle of Duality. As an alternative to a diagram composed of straight lines there is a correlative diagram composed of points, and if three straight lines intersect in a point in the first diagram, the three points in the second will lie on a straight line.

Effecting such a dualistic transformation the three systems of straight lines will now be represented by three systems of points ufi+vgi-I-hl =o ufz+vg2 +h2 =o uf3+vg3 +h3=o forming three scales arranged along a straight line or a curve, according as to whether the straight lines of the correlative system meet in a point or not,' and when three points are taken on these scales whose graduations correspond to three values of z l, z 2, z3 satisfying (r), the three points will lie on a straight line, since the correlative straight lines meet in a point.

Hence to use such a diagram (shown schematically in fig. io) we join any two values of two of the variables on their respective scales by a straight line, and the point of intersection of this straight line with the third scale gives the corresponding value of the third variable.

It will not be necessary actually to draw I / the straight line on the diagram; a piece of lz,? (22) thread stretched across it will give the alignment, or a strip of transparent celluloid, having a straight line engraved down the centre, may be employed for the same purpose.

Given a diagram consisting of straight lines only, representing a three-variable formula in cartesian coordinates, the correlative diagram representing the same formula in parallel coordinates can be constructed geometrically without knowing the analytical expression of the formula represented.

Let D (fig. i I) be a straight line of the left-hand diagram. Take a point M on this straight line whose cartesian coordinates are OH, OK.

The correlative straight line H'K' will be one whose parallel coordinates are AH' = OH, BK' = OK.

Taking in this way the correlative straight lines to any two points on the straight line D, we get by their intersec tion the point P, correlative to the straight line D.

Thus we might take BX', AY', the correlatives of X and Y, the points where the straight line D cuts the axes Ox, Oy, making AX' = OX, BY' = OY.

Proceeding in this way we can replace all the straight lines of the intersection diagram by points.

As an example we have taken the Intersection Nomogram, fig. 6, and constructed from it an Alignment Nomogram, fig. 12.

Suppose, for instance, we want to know the value of R for V =70 m./h., D = r, ioo ft. All that it is necessary to do in fig. 12 is to join 70 on the (V) scale to 1,100 on the (D) scale. This straight line will be found to cut the (R) scale at 1 5% (see transverse line in fig. 12), the required value of R.

Comparing the two figures the advantages of the Alignment Nomogram will be evident. The disadvantages re ferred to in §q. have disappeared, for Fig: 12. there is no tracing back along a line to read its graduation, and any interpolation by eye is only necessary on simple graduated scales.

Proceeding to the direct construction of Alignment Nomograms, without the preliminary construction of an Intersection Nomogram, certain types will now be considered which are particular cases of the general equation (i).

Type

Nomograms with Three Parallel Rectilinear Scales. If the formula to be represented can be put in the form fi +f2 +f = 0 (2) the three systems of points (z 1), (z2), (z 3) can be arranged on three parallel straight lines.

For the systems (z i), (z 2) we take the functional scales u = uifi (3) v= u2 f2 (4) along the two parallel axes Au, B y (fig. 13).

Eliminating f 2 between (2), (3) and (4) gives us for (z3) u2u +ulv +ulu2f 3 - 0 (5).

It is now convenient to revert to cartesian coordinates, taking as origin 0, the midpoint of AB, the axis of x along AB, the axis of y parallel to Au or B y (see fig. 9). Also let OB be denoted by X. With these axes (5) will denote the system of points, u1 +u2' y 14 4- 1 Parallel straight lines of course fulfil this condition and lead to a rectilinear scale as they have a common point at infinity.