2GG BOOK OF MATHEMATICAL PROBLEMS. 



VI. Catenaries. 



1276. An endless heavy chain, of length 2, is passed over 

 a smooth circular cylinder, whose axis is horizontal; c is the 

 length of a portion of the chain, whose weight is equal to the 

 tension at the lowest point, and 2< the angle between the radii 

 drawn to the points where the chain leaves the cylinder; prove that 



TT <i . fir d>\ I 



tan d> + -T f log tan ( T + ) = - . 

 sm< \4 2/ c 



1277. ABCD are four smooth pegs forming a square, AB y CD 

 being horizontal, and an endless uniform inextensible string passes 

 round the four, hanging in two festoons ; prove that 



1 



sin a log cot s sin ft log cot ~ 



& J 



cot a cot ft _l ~ 



log cot ^ log cot ^ 



a, ft being the angles which the tangents at J5, C make with the 

 vertical, I the length of the string, and a the length of a side of 

 the square. 



1278. A heavy uniform chain rests on a rough circular arc 

 whose plane is vertical, the length of the chain being equal to a 

 quadrant of the circle, and one extremity being at the highest 

 point when the chain is on the point of motion; prove that 



1279. A heavy uniform chain rests in limiting equilibrium 

 on a rough cycloidal arc, whose axis is vertical and vertex up- 



