Johnston — Initial Motion. 11 



If we regard in these equations xy% as constant, and x'ij'%' as vari- 

 ables, they are the equations of the axis round which the body begins 

 to rotate when ocyz is fixed. 



So far we have only considered cases where the material system 

 consisted of a single rigid body. If there are several bodies in the 

 system connected together we can divide the forces acting on any 

 body into three sets : the applied forces, the forces due to the con- 

 nections with the other bodies, and the reactions due to any other 

 constraints it may be subject to. Then as before we can find the 

 velocity constituents of the initial motion of each body in terms of the 

 applied forces and the forces due to the connections with the other 

 bodies. This having been done we can obtain, with the aid of the 

 geometrical and dynamical equations derived from the connections, 

 the forces due to the connections, and thence the initial motion of 

 each body and the initial values of the reactions due to any other 

 constraints it may be subject to. 



Take as an example two bodies A and B, a point Q o\i A being 

 connected by a flexible inextensible string to a point P fixed in space, 

 and a point ^ on ^ connected to a point 8 on £ "bj another string. 

 Take the axes of reference such that at the instant of starting the 

 origin coincides with the point Q, and the axis of x with the line QP. 

 Let X, Y, Z, Z, M, N be the constituents of the system of applied 

 forces, and Sw, S«j, hw, Sw^., Sco^, Sw^ the velocity constituents at the 

 time ht from rest of the body A, and x, y, z the co-ordinates of its 

 centre of inertia. We will denote the corresponding quantities for B 

 by the same letters dashed. Let ^, y, % be the co-ordinates of R, 

 x\ y', z' of S, and X, ft, v the direction cosines of RS. Denote by T 

 the initial tension of the string PQ, by T' of the string RS. Also 

 let SFbe the velocity of the point R in the direction RS at the time 

 ht from rest ; then considering the body A we have from equations 

 (1) and (2), since hu = 0, 



m ( - jrSwx + z^oy) = (X + r + T'K) St, 



m {Sv - zScojc + xbooz) = (F + r» Si!, 



w (5m; - xScoy + 2/5a>x) = {Z + T'v) Zt, ... («) 



m (ySw - zSv) + adux - hScoy - ffSooz = {-£ + T' {j/v - Zfj.) } St, 



m ( -xSiv) - hSwx + bSwy -fSwz = {M-\- T'{z\ - xv)} St, 



m {xSv ) - gSwx - fSwy + cSwz = {N + T {xit. - y\) } St, 



fiSv + vSw + [yv - Zfi) 5wj; + (zA. - xv) Swy + {xfi - y\) 5a)» = SV. 



