10S MWCELLANr.' 



17 uniform ' 1 in the form "fa 



_rle rest On a smooth sphere .f radim r; shew that tlio, 



tangent of the inclination of the plane of the triangle t<> the 



horizon is rnr^ t\ where a is the distance of tin- mitres of 



the cireles inscribed in the triangle itself and in tin- triangle 

 formed by j. tinin^ the middle points of the. rods, and p is the 

 radius of the circle inscribed in the triangle. 



18. If a steelyard be construct. -.1 with I 1 whose 



weight 18 inconsiderable compared with that of tin- sliding 

 -ensibility varies inversely as the sum of the 

 sliding weight and the greatest weight which can be 



10. A heavy equilateral triangle hung up on a smooth 

 peg by a string, the ends of which are attached to two of its 

 angular points, rests with one of its sides vertical; shew that 

 the length of the string is double the height of the triangle. 



20. Three equal heavy spheres lying in contact on a hori- 

 zontal plain? are held together by a string which passes round 

 them. A cube, whose weight is II'. i< placed with one of its 

 diagonals vertical so that its lower faces touch the spheres; 



shew that the tension of the string is not less than 



3 \o 



21 . A roof of given span is to be constructed of two beams, 

 which are to be connected at the vertex by a single pin, and 

 the weight of the roof would increase in proportion to the 

 length of the beams; what will be the angle of inclination to 

 the horizon, when the whole pressure on the wall is t Im- 

 possible ? 



Shew that the direction of the line of pressure will then 

 make the same angle with the vertical line which the 1 

 makes with the hori/.ontal line. 



An endless -upports a system of equal heavy 



:hest of which is fixed, th- passing Toond 



every pully and crossing itself between each. If a, /?, 7, &c. 



be the inclinations to the vertical of the successive portions of 



. prove that cos a, cos$, cosy, &c. are in arithmetical 



progression. 



