360 Mis OU8 EXAMPLES. 



tin;:; curve, m.-asuivd along the tangent and n.nn:il 

 at th< 



17. It* a plane pass tin extremity A of the 1 



of a < and be inclined at JIM angle of I.V to 



the piece, 80 cut off will rest in neutral equilibrium, if] 



with 1 .'ii th* vertex of a paraboloid whose latus 



.ths of tin: diann-t'T "f the ba- :nt of 



also at this .-aim- ;r >m J. 



18. A piece of string is fastened at . two 

 fixed points; determine from mechanical consideration 

 form which must be assumed by the string in order that the 

 surface generated by its revolution about the straight line join- 

 ing the fixed points may be the greatest possible. 



MISCELLANEOUS EXAMPLES. 



1. A uniform wire is bent into the form of three sides 

 AB, BC, CD of an equilateral polygon: and its centre of 

 gravity is at the intersection of AC and Bl>. Shew that the 

 polygon must be a regular hexagon. 



2. Three forces act along three straight lines which may 

 be considered aa generating lin< -s in the same system of a 

 hyperboloid of one sheet; shew that if the forces admit 

 Ot a single resultant, it must act along another generating 

 line of the same system. 



ate moves freely about a vertical axis, along which 

 it also slides; while a point in the plane of the gate, and 

 rigidly connected with it, rests on a piven rough inclined 

 plane ; find the limiting position of equilibrium. 



4. Suppose straight lines to be drawn from one of the 

 < litres of the four circles that touch the sides or the 

 produced of a given triangle to the other three cent:-----, and 

 let these straight lines represent three forces in magnitude and 

 direction; then the straight line joining the first centre with 

 the centre of the circle circumscribing the triangle will re- 

 present in magnitude and direction one-fourth of the resultant. 



