Figure 16

As we can see, the result of applying the Hamilton’s diagram on the plane of the space of phases, consists, in practical terms, in drawing a cross over this latter and in doing so the surface is divided into 6 areas:

• A+ . Named in this way because there are only Pushing ups (A), that is, functions which adjust rising oriented graphs and moreover, this area is on the positive side of the phase plane.
• A-. Named from the existence of Pushing ups (A) but the graph is situated on the negative side of the phase plane.
• AB+. Named from the existence of either Pushing up and Pulling down functions. Moreover, it is situated on the positive side of the phase plane.
• AB-. The same as above but this area is located on the negative side of the phase plane.
• B+. Area where there are only Pulling down functions (B), functions which adjust falling oriented graphs and this area is located on the positive side of phase plane. B-.The same as above, but this area is situated on the negative side of phase plane.

In view that an adjusting function of a graph, could be situated on the zero line which divides the positive from the negative side of the plane (something which effectively happens in the cases of free rise and free fall both situated on the left hand upper end and the right hand lower end), the following two divisions must be added to the above:

• 0+. It corresponds to the zero line stretch from the centre to the upper left hand corner of the phase plane.
• 0-. It corresponds to the zero line stretch from the centre to the lower right hand corner of the phase plane.

The method used to inform of the position which a given graph occupies over the "synthesis diagram", is based on a system of angular coordinates of position (two angles expressed in sexagesimal system) which in turn represent the angles of turn of each of the two fundamental vectors acting over the graph subject of analysis, and that they always are the degrees of Pushing up Force and the degrees of the Pulling down Force.

In the case of a graph representing equity quotations, the degrees of Pushing up Force represent degrees of Demand and the degrees of the Pulling down Force represent degrees of Supply. But if graphs depicts cash-flow, degrees of Pushing up Force will stand for degrees of efforts to be made to get revenues and the Pulling down Force degrees will represent the degrees of effort to reduce costs.

The subject of this section is to explain why the angular coordinates indicate the point (the position) on the diagram where the adjusting function is situated, and quantify at the same time, the Pushing up and Pulling down Forces acting on the graph.

Figure 17  

Notice that the two facing systems of coordinates, are not directly applied on the diagram, if not slightly apart. This is to avoid the zero value of both lower left and upper right corners. Instead, their value are 0,25 degrees on both axis. On the other hand, the value of the other two corners (upper left and lower right) are both 180º.

Under these conditions, the Edgeworth’s Box axis system is as follows:

The coordinates of Edgeworth’s Box, emulate numerically the turn settlements of vectors of Hamilton’s diagram. Indeed:

The maximum point of free rise, which in Hamilton’s Diagram is represented in the upper left hand corner, with both vectors pointing up, which in an equity chart means the maximum Demand (first vector) driving the graph to rise from demand side, and the minimum Supply (second vector) driving the graph to rise from supply side.

In the Edgeworth’s Box the same maximum point of free rise is numerically represented as 180 / 0.25 which means the maximum Demand (180º) and the minimum Supply (0,25º) and this point in Edgeworth’s Box is situated in the same position in Hamilton’s Diagram.

In Edgeworth’s Box the same minimum point of free fall is numerically represented as 0,25 / 180 which means the minimum Demand (0,25º) and the maximum Supply (180º) and this point in Edgeworth’s Box is situated in the same position as in Hamilton’s Diagram.

Let us now see an example of how the mathematical - IT system of the Symbol works:

We have a curve of quotations, which for the IT System is a series of data values. From it the system works in the following way:

The technique is quite simple, as it is only a matter of changing the maximum value of the series to the maximum value of the pattern scale and the minimum value of the series to the minimum value of the pattern scale and then, taking this change into account, transforming the intermediate values of the series into intermediate values of the pattern scale.

The final result is that the original values of the series have been transformed into others susceptible to fit them into the adjusting functions of the system, but keeping the exact form of the original graph.

Recall that the system analyses forms of graphs, not their numeric values from which graphs are made from. Hence, if two graphs are exactly the same, but their numeric values differ, the system would tell us exactly the same thing.

Indeed: If somebody tries to copy a similar system but without having an IT processing system, they assign, at a rough guess, degrees of the Pushing up and Pulling down forces present in a graph.

If a user company sends determined data to be processed and gets the degrees of forces resulting from the process, it can assess over the time if what it got was correct or a fraud. For this, it must only send data, apparently similar, but which in reality all values are multiplied by a linear factor (i.e. 1.637, 21. or 5).
This is so because the curve would be the same, even though data is different, and for the system (form processor) the result would be the same as the original.

B / Once the system has transformed the numeric values of the graph to pattern values, it explores the 516.961 functions which possesses and detects which of them adjusts better to the pattern data. With each function there’s a record containing the coordinates of the allocation in Edgeworth’s Box. It should only show these coordinate values to know the position where this graph is situated.

We habitually call the Symbol (S) to the mathematical - IT system subject of this survey, which we already know is capable of finding the two fundamental forces which act upon the last point of a determined graph, and we call Diagram of the Symbol (SS), a diagram contained in Edgeworth’s Box, that is Hamilton’s diagram but affected by a dividing diagonal.

In this section we will deal with the SS most common singular points, where we can say a graph is about to start to rise or to fall, taking into account that the first figure will express the magnitude in degrees of the Pushing up Force and the second figure the magnitude in degrees of the Pulling down Force, and inside the parenthesis there is the Hamilton’s area, where the adjusting curve is situated. These points are the following:

• 180 / 0.25 (0+). Graph is in "free rise" and this situation can not last long. It will fall soon.

• 0,25 / 180 (0-). Graph is in "free fall" and this situation can not last long. It will rise soon.

• 0,25 / 0,25 (AB+). Graph has touched the line which marks its rising tendency. If the first figure increases and becomes higher than 0,25 it will bounce back on this tendency with a Pushing up impulse. If, on the contrary, it is the second figure which rises, the rising tendency will result perforated, and graph will suffer a Pulling down impulse.

• 0,25 / 0,25 (AB-). Graph has touched the line which marks its rising tendency. If the first figure increases and becomes higher than 0,25 it will bounce back on this tendency with a Pushing up impulse, but if it is the second figure which goes up, graph will hit the falling tendency and it will suffer a Pulling down impulse.

• 0,25 / 90 (AB+). Graph is on the verge of a qualitative change of its behaviour which consisted in going down to get impulse and rising more. If the first figure goes up it will get a Pushing up impulse and it will for the moment maintain this behaviour. Yet, if it is the second figure which rises, the graph will change qualitatively and its new behaviour will consist in rising to get impulse and fall further down.

• 90 / 0,25 (AB-). Graph is on the verge of a qualitative change of its behaviour which consists in rising to take impulse and then fall further down. If the first figure increases it will qualitatively change and its new behaviour will be falling to take impulse and rise higher. But if is the second figure which rises it will get a Pulling down impulse and will keep on behaving in the same manner.

Naturally, the points situated between those above will show intermediate characteristics. It should be pointed out, however, that if the projection of a graph over the SS evolves rising on its left hand side (AB+), enters into (A+) and then moves into (A-) this means that graph exhausts its Pushing up force. It is also necessary to highlight that if the projection of a graph over the SS evolves going down on the right hand side (AB-), enters into (B-) and then penetrates into (B+), this means that the graph is powered to go down.