(4) Lastly, suppose we wish to determine P and J! and 

 the position of equilibrium. 



Then w- must give the beam tlic nmst mend dis]>' 

 possible in the plane of the forces; let AA' c. 



let a be the angle which A A makes with the vortical. 

 conceive the beam brought into its second position l.y t\v. 

 steps ; first let it be moved parallel to itself till tln> lower end 

 s to A\ and next let it revolve round A' through an 

 angle 80. The displacement of A estimated along the line 

 of action of E is csm a. The displacement of G cstin 

 vertically downwards is 



aS0 sin 6 c cos a. 

 The displacement of B along the line of action of P is 



c cos (a + j - e\ - 1) cosec 080, 



that is, c sin (0 - a) b cosec 0S0. 



Therefore by the principle of virtual velocities 

 JF(aS0sin0-ccosa) 4-jRcsina 



+ P{c sin (0 - a) - & cosec 050) = 0; 

 that is, 



( JFa sin 0-Pl cosec 0) 80 + (P sin - TF) c cos a 

 + (#-Pcos0)csina = 0, 



and 80, c cos a, and c sin a are independent ; therefore 



Wa sin - PI cosec = 0, Psin - TF= 0, # - Pcos = 0. 



