">l5 'POSITIONS RESPECTING 



lilirium of the body or bodies 



Beginners often fall into error by OJM /////'// im-Mi-rect 



for the tensions of strings and the mutual forces 



bodies in contact, instead .mining tin- cnvet values 



from die equations of equilibrium. 



We will give here two propositions, respecting forces acting 

 in a plane, which involve important results. 



I . Forces act at the middle points of the sides of a 

 polygon in the plane of the polygon; the fore, 

 angles to the sides, ami ; -lively ir.jM.rtiMual i 



sides in magnitude: shew that the forces will In- in (.juili- 

 brium if they all act inwards or all act outwards. 



The result here enunciated has been already shewn to be 

 true in the case of a triangle; see the Proposition IV. at the 

 end of Chapter n.; the general proposition is obtained liy 

 an inductive method. 



Suppose for example that the proposition were known i 

 true for a four-sided figure ,; 



then we can shew that it must be 

 true for a five-si.';..! li-uic. Let 

 ABCDI, he a five-sided figure; and 

 let forces act at the middle points of 



.-ides in the plane of tl, 

 at right angles to the sides and 

 spectively proportional to the .-ides 



in magnitude: suppose for the sake of distinctness that the 



forces all act outwards. 



Join AD. By hypothesis a certain system of forces a< 

 outwards on the four-sided figure ABC J ! ! in equili- 



brium: and from this it follows that the assigned forces acting 

 on AB, BC, CD must be equivalent to a single f.-ree aeting 

 at the middle point of AD, towards the ii. the four- 



sided figure ABCD, proportional to AD in magnitude. 



Also the assigned forces acting on DE, KA must in like 

 cr be equivalent to a single force acting at the middle 

 of AD, towards the inside of the triangle AED, pro- 

 portional to AD in magnit 



