EXAMPLES 89 



19. A string of weight W is suspended from two points at the same 

 level, and a weight W is attached to its lowest point. If a, j8 are now the 

 inclinations to the vertical of the tangents at the highest and lowest points, 

 prove that tang W 



tan/3 W 



20. A heavy string of length I is supported from two points, and at 

 these points the string makes angles a, ft with the vertical. Show that 

 the height of one point above the other is 



I cos !(<* + 0) sec I (a - /S). 



21. Prove that the direction of the least force required to draw a 

 carriage is inclined at an angle 6 to the ground, where asm 6 = bsine, 

 a and b being the radii of wheels and axles respectively, and e being the 

 angle of friction. 



