432 Proceedings of the Royal Irish Acadeiny. 



problems in the dynamics of a rigid body wbicb has freedom of the tbird 

 order. 



Let an impulsive icrench act ujjon a quiescent rigid hody; it is required 

 to determine the instantaneous screio about ivMch the hody will commence to 

 twist. 



It can be easily shown {Screios, p. 59) that the impulsive wrench, 

 ivherever situated, can always be adequately represented by an impul- 

 sive wrench on a screw of the three-system. The problem is therefore 

 reduced to the determination of the point corresponding to the instan- 

 taneous screw, where that corresponding to the impulsive screw is 

 known. In the special case where the freedom degrades to the rota- 

 tion around a point, the problem now before us reduces to that solved 

 in Poinsot's celebrated memoir. 



"We have first to draw the conic of which the equation is {Screws , 

 p. 133) 



«i2 e^- + wo2 ei + ui ^3' = 0. 



This conic is of course imaginary, being in fact the locus of screws 

 about which, if the body were twisting with the unit of twist velocity, 

 the kinetic energy would nevertheless be zero. If two points 6, (ft are 

 conjugate with respect to this conic, then 



tci^ 61 ^1 + %- 6i (jio + u-i- 63 <j!)3 = 0. 



The screws corresponding to and (f) are then what we have called 

 conjugate screivs of inertia. 



This conic is referred to a self-conjugate triangle, the vertices of 

 which are three conjugate screws of inertia. It is possible to find one 

 self-conjugate triangle to the zero-pitch conic, and to the conic of 

 inertia just considered. The vertices of this triangle are of especial 

 interest. Each pair of them correspond to a pair of screws which are 

 reciprocal, as well as being conjugate screws of inertia. They are 

 therefore what we have designated as the j!?rmcj}j«? screws of inertia 

 {Screivs, chapter YI.). They degenerate to the principal axes of the 

 body when the freedom degenerates to the special case of the rotation 

 around a fixed point. 



When referred to this self-conjugate triangle, the relation between 

 the impulsive point and the corresponding instantaneous point can be 

 expressed with great simplicity. Thus the impulsive point ^, whose 

 co-ordinates are 



Oi ^ii ^ Pa. ; ^2 M2^ ^ ]??. ; ^3 u-i -^ p^, 



corresponds to the instantaneous poiat whose co-ordinates are d)_, do, 6^. 

 The geometrical construction is extremely simple when derived from 

 the theorem thus stated. 



If (j> denote an impulsive screio, and 6 denote the corresponding 

 instantaneous screw, then the polar of ^ with regard to the zero-pitch 



