228 KANSAS UNIVERSITY QUARTERLY. 
Case (a) Continued. 
It is evident that another series of catenarys can be drawn 
through the two abutments B, and B, of Fig. 9 ina different way 
from those of the class considered. We may have a series of 
curves of varying parameter through B, and B,, the axis of which 
are to the left of B,, and which consequently do not have their 
lowest points, or vertices, within the limits of the figure. 
If analyzed independently it will be found that they may be 
treated in the same way as those with axis between B, and B,, as 
would be supposed. They are indeed no different in any way 
from the other class. Simply, as the cord is made shorter the ca- 
tenary becomes flatter, removing the axis nearer to the lower abut- 
ment, until when acertain length is reached the axis is at the lower 
abutment. Then with the shortening of the cord the axis moves 
farther beyond the lowerabutment, until finally when the length is too 
short to reach from one abutment to the other, that is less than the 
straight line B,B,, the construction fails altogether, as it should. 
An inspection of Figs. 7 and g will show this final limit very 
satisfactorily. Looking at Fig. 7 it is plain that the line OB will 
become tangent to the s-curve at some limit as Z XOB decreases. 
The limit for XOB, below which the construction fails, can be seen 
: ‘ee , fds 
at once by differentiating the s-curve and making x in (=) equal 
dx 
zero. This shows that at xo, that is at point O, the angle of the 
tangent to the curve is 45°. Now notice that this fixes the limit 
for the length of the cord just where it would be found from a con- 
sideration of the physical conditions of the problem, as shown in 
Fig. 9. 
If Z XOB is 45° at the limit, then there 
BM=OM. 
for this case, 
| ]2_vy2—D; 
a [= Dip ve, : 
V D?-+-v? is the length of a straight line from B, to B,, Fig. 9, 
evidently the extreme minimum limit for the length of cord. 
The length of the cord which brings the axis just at the line of 
the lower abutment can be seen thus: 
Substitute in equation of the catenary the values from Fig. g and 
c D D 
ea ca) 
————— 
ee ee? ee ee 
