537] 
SOLUTIONS OF A SMITH’S PRIZE PAPER FOR 1871. 
511 
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10. An endless heavy chain of given length is suspended from two fixed points in the 
same horizontal plane: show that (.subject to a condition as to the length) the figure of 
equilibrium may consist of portions of two distinct catenaries. 
The two parts of the chain will each of them be a portion of a catenary, viz. 
they will either coincide with each other, forming a twice repeated portion of a 
catenary (which is always a possible position of equilibrium), or they will form portions 
of two distinct catenaries. That the latter form is in some cases possible, appears 
from the case of a very long chain. It is then clear that there is a position of 
equilibrium in which the upper catenary is nearly a straight line. It may be added, 
that, as the length of the chain diminishes, the two distinct catenaries approach more 
and more, and for a certain value of the length become coincident; for any smaller 
value of the length, the only position is that consisting of a twice repeated portion 
of a catenary. But to obtain the solution in a regular manner, observe that, in order 
to the existence of such a form of equilibrium, the necessary condition is, that the 
tension at A (or B) must be equal in the two catenaries. Now the tension at any 
point of a catenary is proportional to the height above the directrix of the catenary; 
hence the condition is, that there shall be through the points A, B two catenaries having 
the same directrix, and such that the sum of the lengths is equal to the given length 
of the chain. 
Take AB = 2a, the length of the chain = 21. Take /3 for the distance of the 
directrix below the points A, B; c for the parameter of the catenary (or distance of 
its lowest point above the directrix), /3, c being of course unknown. Then taking the 
origin at the mid-point of the directrix, and the axis of y vertically upwards, the 
equation of the catenary is 
whence for the point A or B, 
a a 
/3 = l(e-° + e~°), 
and the arc measured from the lowest point is 
Hence, assuming that there are two distinct catenaries, if the parameters are c, c', we 
have 
a 
a 
a 
a 
%c(e c + e c ) = \c' (e c ' + e c '), 
a 
a 
a a 
%c (e c — e c ) + \c' (e 
which are the conditions for the determination of c, c' ; and it is to be shown that 
these can be satisfied otherwise than by taking c = c'.