222 KANSAS UNIVERSITY QUARTERLY. 
Divide OD into an odd number of small equal divisions, say ap- 
wy 
: I 3 : 
proximately —-. Then draw vertical lines through every other 
IO x 
division point, as shown in the figure, Fig. 6, so that OY will come 
in the middle of a space between verticals. Draw a straight line 
through D and O’. Then draw through a and‘1, b and 2, c and 3, 
etc. The resulting series of lines then envelopes the curve, which 
will be found determined with accuracy without the necessity of 
drawing a curved line, if the spaces between verticals be made 
small. The figure will make clear that the curve touching each of 
these straight lines goes through point O’ and that the condition of 
I 
a constant sub-tangent equal —, or c, obtains throughout, and 
m 
hence that the curve is correctly and satisfactorily drawn. 
Having now a simple construction for this exponential curve, 
and understanding the derivation of the catenary from it, we may 
easily construct a catenary of any desired parameter; and are, 
therefore, ready to pass to the consideration of the question of 
adapting this method to more general cases of the catenary con- 
forming to given sets of conditions as to points, length of cord, ete. 
CONSTRUCTIONS FOR THE CATENARY CONFORMING TO CIVEN 
CONDITIONS. 
Aside from the case already noticed of plotting a catenary with a 
given or assumed parameter, we have from purely physical consid- 
erations the following important cases: 
CASE I.—SUPPORTS THE SAME HEIGHT. GIVEN. 
(a) Given a certain length of cord greater than the distance be- 
tween supports, required the levei to which the curve will fall and 
a plotting of the catenary formed. 
(b) Given the level to which the curve is to fall, required the 
length of cord and a plotting of the catenary. 
CASE, 11.——-SUPPORTS AT DIFFERENT LEVELS: 
(a) Same as under I. 
(b) Same as under I. 
Remarks: It should be noticed that the weight per unit length 
of the cord does not affect the form of the curve, but does deter- 
mine the tension in the cord. 
The tension of the cord on the abutment or at any. point, total 
weight of cord, etc., can, of course, all be readily determined graph- 
ically for each of the cases above mentioned, when once the curve 
is plotted, by means of the properties already considered, so these 
features will not be treated again in the discussion of these cases. 
