220 KANSAS UNIVERSITY QUARTERLY. 
That is, s is the difference of ordinates at each point, as shown 
in Fig. 4. These considerations show then that if a satisfactory 
drawing-board construction can be found for this exponential curve 
a means is already at hand for plotting the catenary, at least for the 
simple case of a given parameter. 
To discover the desired kind of construction for this exponential 
curve—one which will admit of the curve being drawn with as 
much facility as the parabola or ellipse—necessitates leaving the 
question of the catenary itself for a time to investigate the proper- 
ties of this exponential curve. 
CONSTRUCTION FOR THE EXPONENTIAL CURVE. 
The perfectly, general form of this curve 1s 
mx 
y=ae (19) 
Its most notable property is that its sub-tangent is constant. 
Foe Sateen Cast Ea 
Fig. 5. 
That is, DM, Fig. 5, is the same for a given curve wherever the 
point P may be chosen. For 
PM eee: ay == BU) h0l Eten 
ow pr-(as)= a cents 
