1<, FORCES ON A PARTiri.i:. 



Muli> :irst by cos a', the second by CMS/?, and the third 



ami u<M. rhen,if0 t , 6L... denote the ftnglei \\hieh 



'... make with the arbitrarily drawn straight line, and 

 tf the anirle wliich the resultant Ji makes with it, we ha\ 

 the formula quoted in Art. 28 for the cosine of the ang! 



i two straight lines 



Ji cos B = P l cos 0, + P f cos 0, 4- 



30. From Art. 20 it is obvious that a given force may 

 be resolved into two others in an infinite numher of ways. 

 When we speak of the resolved part of a force in a 

 direction, as in the preceding Article, we shall <////</>/ 



unless the contrary is expressed, that the 



1 into two forces, one in the /// and the 



other in a directional right angles to /// The 



former component we shall call the resolved force in the 

 direction. 



AVhen forces act on a particle it will be in equilibrium, 

 provided the sums of the forces resolved al three 



lions not lying in one plane are zero. Km- ii the forces 

 do not balance, they mint have a single resultant; and as 

 a straight line cannot be at ri_cht angles to three 

 lines which meet at a point and are not in the same plane, the 

 resolved part of the resultant, and therefore the sum of the 

 resolved parts of the given forces, along these three sti 

 lines, could not vanish, which is contrary to the hypothesis. 



31. In Art. 26 we resolved each force of a system 

 three others along three rectangular axes. In the same way 

 we may, if we please, resolve each force along three straight 

 lines forming a system of oblique axes. For whether the 

 figure in Art 24 represent an oblique or red il. de- 

 piped, the force AF may be resolved into Ah '. and 

 the latter again resolved into AH and AC. \ 



Fultant of a system of force- resented by \ 



nal of an oblique parallelepiped, and for equilibrium it 

 be necessary that this diagonal should vanish, and therefore 

 that the edges of the parallelepiped should vani.-h. 



The following three articles are particular cases of the 

 equilibrium of a particle. 



