114

and <p being found, c, t , and a? may be deter-mined byCase III.

case xv.

When x and y are given, to find the rest.

Log.tang,(45°-[-^) _ y

sec.$ver.sin.<|> mx

from which <p

may be found by approximation; andwhen <p is found, c, t, and z may be deter-mined, as in Case III.

10. Since it has been shown in Art. 8.that y '=~ and, in receding from the ver-tex, is always positive, y must also be a

positive quantity; consequently y must al-ways increase, and the curve must there-fore continually recede from its axis.

11. From the same equation it evidently

. i . , c % x y i

appears that y oc — cc — or — oc —

r r %/ Z Z X z 9

and since z is always a variable quantity,~ must also be variable, and consequently

the catenary has no point of contrary flex-ture; and since the curve is always con-cave towards its axis, it follows also that,in receding from the vertex, the ratio of yto a? continually diminishes.