In Figures 7(a) and 8(a), for the single-orifice case, the velocity and pressure contours are almost symmetrical at the X-Y plane and X-Z plane, whereas, for the multiorifice case, with the help of auxiliary gas orifices and fan gas orifices, the spray pattern becomes totally different.

(a) Contours on the X-Y plane (left: calculation; right: experiment); (b) contours on the X-Z plane (left: calculation; right: experiment).

(a) Pressure contours for single-hole nozzle: X-Y plane (left); X-Z plane (right).

Therefore, the springback process is divided into horizontal direction along the x-y plane and vertical direction along the x-z plane, as shown in Figure 9, where Aa represents the total springback value, Aay represents the springback value along the x-y plane, and Aaz represents the springback value along the x-z plane.

This is because the aluminum profile has been bent horizontally along the x-y plane before the vertical bending.

The theoretical curve of the horizontal line along the x-y plane is as follows:

Also, by virtue of the symmetry present in both surfaces A and B, and the way these symmetry planes intersect each other at right angles, it is evident why the complex conjugate roots have to occur in pairs equi-distant from the G-axis, i.e., behind and in front of the original Cartesian x-y plane! Returning to Figure 2, it should now be clear that the two, somewhat tentatively placed, complex conjugate roots are indeed located at x = -0.5, but occur a distance 0.866 units behind and in front of the graph as it is currently plotted in two-dimensions.

Finally, a section taken at H = 0 through surface A will reveal the original octavic equation as it would appear in the Cartesian x-y plane.

This paper has provided the full and complete visual connection between the simple polynomial y = [x.sup.n] - 1 presented in the Cartesian x-y plane, and its roots, which are invariably presented in the complex Argand plane.

However, for the bending stress in the X-Y plane, there is a large dissimilarity between the two different loading models, as shown in Figure 13.

That is, the superimposed stress in the X-Z plane is the sum of the axial direct stress and the bending stress in the X-Z plane, and the superimposed stress in the X-Y plane is the sum of the axial direct stress and the bending stress in the X-Y plane.

A plot in the Cartesian x-y plane reveals a parabolic curve with its vertex located at

A sample calculation, taken for H values either side of the original two-dimensional x-y plane, is given in Table 1 to help the reader understand how the surface A is calculated.