78 PROPOSITIONS RESPECTING 



I. 1' TOGS act at the an-ular joints of a t. n in 



ular t. thr i.p|,..~itr 1-ir, 



proportional t> the areas of the faces in magnitude : slu \\ that 

 the forces will be in equilibrium. 



Let ABCD represent th- 

 dron. 



(1) Resolve the forces parallel 



/'. Let p denote the ] 

 dicular from A on the fan- //' 7> : 

 then the resolved part of the 



at A is J' x area of BC1>, that is, 



3 volume of tetrahedron 

 AB 



We obtain the same expression for the resolved part of the 

 force at B. The forces at C and D have no resolved part 

 parallel to AB. Thus the forces resolved parallel to All 

 vanish. 



(2j Take moments round AB. Let q denote the perpen- 

 dicular from C on the straight line AB\ 6 the angle Le- 

 the planes BAD and BAC. Then the moment of the force 

 at C is q cos 6. area of ABD, that is, 



qAB cos e . area of ABD 2 cos . area of ABC, area ofABD 

 AJf ~ 



We obtain the same expression for the moment of the 

 force at D. Thus the moments round AB vanish. 



'e these results hold for any edge of the tetrahedron 

 the forces must be in equilibrium. 



II. Four forces act on a tetrahedron at riirht an cries to 



laces and proportional to their areas, the points of 



Application of the forces being the centres of the circles cir- 



cumscribing the faces: shew that if the forces all act in- 



wards or all act outwards they will be in equilibrium. 



i his case the forces all pass through a point, namely 

 the centre of the sphere described round the tetrahedron. 



