Johnston — Initial Motion, 3 



From equations (2) we can find Sw, hv, Stv, 8(o^, Sw^, Su>,, and then 

 by means of equation (1) 8x, Sy? ^2, and consequently get tlie initial 

 direction of motion of any point. Wlien the origin is taken coinciding 

 with the centre of inertia equations (2) become 



XSi = jnSu, YSt = mdv, ZU = m^w \ 

 IiSt = aSojx — h^ooy — ffScoz 

 MSt = - /iStur + bSooy -fSuz 

 N^St = — ffBcox —fSaiy + cScoz 



(3) 



and when the axes are the principal axes through the centre of 

 inertia, 



XSt = mSu, TSt = mSv, ZSt = mSw \ 



(4) 



Ldt = AScox, MSt = BScoy, mt = CScoz ) 



where A, B, C are the principal moments of inertia. 



CONSTEAINED MoTION. 



When the motion of the body is constrained, at the moment of 

 starting from rest under the action of the system of applied forces, 

 there will be a system of forces of reaction called into play. Taking 

 as axes of reference axes fixed in space which coincide with the 

 principal axes of the body through the centre of inertia at the moment 

 of starting, we have from equations (4), if X, Y, Z, Z, If, JV, 

 Xi, Yi, Zi, Li, Ml, iVi are the constituents of the system of applied 

 forces and the forces of reaction respectively, 



mSu = {X + Xi) St, mSv = ( F + Fi) St, mSiv = {Z + Zi) St. (5) 

 ASccx ={Z + Li) St, BSaiy = (if + Mx) St, CSw^ = (N + iVi) St. (6) 



From equations (1) we see that any geometrical consti-aint of the 

 point xyz gives a linear relation connecting the six quantities Su, 8v, 8w, 

 So)^, Soiy, 8w.. Hence if there are six independent geometrical con- 

 straints we get six linear equations connecting these six components 

 of velocity, and consequently each must be zero, and therefore 



Z + Zi = 0, F+Fi = 0, Z+Zi = 0, 

 L + Zi = 0, M+ My = 0, N+ Ni = 0, 



for 8t though small does not vanish, which show that in this case the 

 initial reactions must balance the system of applied forces, a fact we 

 know from a priori considerations. 



