10 Proceedings of the Royal Irish Academy. 



The geometrical equations (1) reduce to 



Soij! 5a)y Scos 

 o: ~ y ~ z 

 Therefore 



J + Zi _ M+Mx _ N+Ni 

 Ax By Cz 



Since we suppose the constraint at the fixed point smooth, 



Lx=yZi-zYx = -{yZ-zY), 

 Ml = zXi -xZi=- {zX - xZ), 

 Ni = xYi-yXi^-{zY-yX). 



Therefore the point xy% may be any point on the common curve of 

 intersection of the system of hyperboloids 



L-yZ+zY _ M-zX + xZ _ N - xY + yX 

 'Ax ~ 1y ~ Cz , 



Each pair of these three hyperboloids has a common generator which 

 is not a generator of the remaining one ; consequently the locus is a 

 tested cubic. 



The corresponding theorem for impulsive force is : — If a body at 

 rest is acted upon by a system of impulsive forces whose constituents 

 referred to the principal axes through the centre of inertia are X, Y, Z, 

 L, M, iV", the instantaneous axis of rotation will pass thi'ough the 

 centre of inertia, if any point on the twisted cubic 



Z-yZ+zY 3I-zX + xZ N - xY + yX 



Ax By Cz 



is fixed. 



There is a more general question : to fijid the locus of the point 

 P which must be fixed in order that the initial motion may be a 

 rotation round a line passing through a given point Q in the body. 

 If we take the same axes as before, and let ^, y, % be the co-ordinates 

 of P; x'^ y', z' of Q, we find, on proceeding in the same manner, 

 that the locus is the common curve of intersection of the cubic 

 surfaces 



A{x — x') +m{x{yy' + zz') —x'{y'^ + z-)} B(y — y') +m{y{xx' + zz') —y'{z^+z'^)} 

 L-yZ-i-zY " M-zX+xZ 



_ C(z - z) + m{z{xx' -\-yy') - z'(a;- + y^) } 

 N-xY-VyX ' 



which is a curve of the seventh degree. 



