8 Proceedings of the Royal Irish Academy. 



The initial position of the acceleration centre is the intersection of the 

 planes 



( M N\ ^ 



t L M\ 

 ^ + mf t/— -a; — I = 0. 



These planes have a common line of intersection, if 



■which is consequently the condition that the initial motion should be 

 a pure rotation. 



The principal axes through the centre of inertia are not always the 

 most convenient. Consider the case where the constraint is formed by 

 a fixed smooth circular cylinder of very small radius passing through 

 a tube in the body which it fits. We will take our axes of reference, 

 such that the axis of x coincides with the axis of the cylinder, and the 

 axis of y passes through the centre of inertia. Since the body is free 

 to move along the cylinder and rotate round it, Xi = 0, Xi - 0. And 

 as these are the only motions it can have, hv = 0, ^w = 0, Swj, = 0, 

 ^w, = 0. Therefore, from equation (2), remembering ^ = 0, z - 0, 



mdu = X5t, 



0={Y+ Yi)St, 

 mpSux = {Z+ Z\) St, 



aSux = I/St, 

 - hSwx = {M + Ml) 5t, 

 - ffSwx - mySu = (iV+ iVi) U. 



Therefore the constituents of reaction are 



Zi = 0, Yi = -Y, Zi = -Z+'^L; 



a 



Zi = 0, Mx = -M--L, Ni = -N-^L-yX. 

 a a 



The initial direction of motion of any point xy% is found from the 



equations 



X 



Si" = — 5<, 



m 



Sy = --Z5L 

 l~ = -LU. 



