Johnston— /w^Y^■a/ Motion. 



13 



face. Also let li, mi, n^ be the direction cosines of the normal to the 

 rough surface at this point, and Zj, mj, n^, Iz, nis, Us the direction cosines 

 of any pair of right lines at right angles to limiUi and to each other. 



Denote by ^i, R^, Hz the initial reactions at this point in the 

 directions of these lines. Then, adopting the same notation as on 

 page 5, except that everywhere we replace a^i^iZi, x^y^z^, Xzyz^z by 

 xy%, we have, if Swjj 8% denote the components of velocity of xyz in 

 the directions l^m^n^, 4»«3%, 



Pl + i?l(l+\ll) + ^2Al2 +RzMz =0. 



{Pi + Ri A.12 + i22 (1 + A22) + -R3 A.23} ^t = mSu2. 



{P3+-R1A13 +-R2A13 + Rz {I + \z3)} St = tnSuz, 



Since the point xyz begins to move in the same right line as the 

 force of friction acts, we must have 



SM2 



R^ 



Rz 



Let each of these quantities equal - ht, then 



mluz = kRiSt, 

 mduz = kRzSt. 



Since 8t is essentially positive, k must be negative, for the force of 

 friction is in a direction opposite to that in which the point begins to 

 move. 



We have now 



Fi + Ri{l + \n) + R2\n + RzMz =0, 



Ri + Ri^n + R2{l-k + \22) + R3\23 = 0, 



F3 + Ri\i3 +R2M3 +R3{l-k + \zz) = 0, 



and if motion takes place, 



^,.■'R^■' = R2^' + R3^ 



where /a is the coefficient of friction. Therefore 



Pi A.12 A 13 



P2 l-k+\22 Ms 



Pz A23 1— 7i' + A33 



Pi 1+All Al3 



P2 A12 A23 



P3 A13 l-^ + A33 



Pi 1+Aii A12 

 P2 A12 1-A;+A22 



P3 Ai3 A23 



■ (9) 



Physical considerations tend to show that this equation can only 

 have one negative root if the point xyz has an initial motion ; for, if it 

 had more than one, more than one initial motion would be possible ; 



