If weights be suspended from various points of a hang ing chain, the intervening por tions will form arcs of equal ~ **catenaries**, since the horizontal tension (wa) is the same for all.

Again, if a chain pass over a perfectly smooth peg, the **catenaries** in which it hangs on the two sides, though usually of different parameters, wifi have the same directrix, since by (10) y is the same for both at the peg.

Hence for every tension greater than the minimum tension there are two **catenaries** passing through A and B.

Since the tension is measured by the height above the directrix these two **catenaries** have the same directrix.

Now let us consider the surfaces of revolution formed by this system of **catenaries** revolving about the directrix of the two **catenaries** of equal tension.

14) be two **catenaries** having the same directrix and intersecting in A and B.

Draw Pp and Qq touching both **catenaries**, Pp and Qq will intersect at T, a point in the directrix; for since any **catenary** with its directrix is a similar figure to any other **catenary** with its directrix, if the directrix of the one coincides with that of the other the centre of similitude must lie on the common directrix.