230 KANSAS UNIVERSITY QUARTERLY. 
to obtain NHK, the other half of the curve, only plotting so much 
as will evidently be needed. This can be done very readily. 
Now from a study of Fig. 8, and the steps there taken it can be 
plainly seen that the position for the Y-axis, in the case represented 
in Fig. 10, must be suchas to make possible the diagram of straight 
lines shown here in this figure. 
The portion of the figure on the one side of OY is merely the 
construction of Fig. 8, applied to the conditions of this half of the 
curve, and the portion on the other side is the same construction 
applied to the conditions of that half. 
The values determined for c from the two portions of the dia- 
gram, must, of course, be the same here, so Mz and Nd must meet 
OY inthe common point t. That is, the vertical axis OY of the 
required catenary must be so located that lines from a and b, points 
determined by it, to N and M respectively will give intersecting 
points, H and F, properly situated, so that the points d and z, de- 
pendent upon them, will give Nd and Mz, intersecting at the point 
t on the axis. Then Ot is the constant c, and may be laid off at 
once downward from O on OY and the required curve plotted. 
It is plain that all of the quantities of this diagram are so involved 
in each other that the position of OY fulfilling the necessary con- 
ditions cannot be located at once definitely. But when the two 
halves of the simple catenary have been drawn at their proper dis- 
tance, D, apart, it is possible to see within a very small limit of 
error just where OY should be for the fulfillment of these condi- 
tions. Estimate its position closely thus and draw the diagram, 
when it will be found that Mz and Nd do not quite meet in the 
same pointon OY. A slight change of the position, now, will bring 
the two intersections on OY much nearer to coincidence and a very 
few such trials will locate OY to a degree of accuracy equal to that 
which could be attained by any method. 
Conclusion, 
All of the cases which would actually be encountered in the 
plotting of the curve of a material’ cord of uniform weight have 
now been considered and solved entirely by graphical steps. The 
rules outlined under. each case, having been demonstrated, may 
hereafter be taken as satisfactorily established and a catenary plot- 
ted to conform to any set of conditions which may be physically 
possible, with no computing whatever. 
