34 Proceedings of the Royal Irish Academy. 



must cut the circle in Q, -which coiTespouds with the requii'ed impul- 

 sive screw. We may state the matter also in a somewhat different 

 manner. Let S be the vertex of a variable triangle inscribed in the 

 circle whose two sides SF and SQ pass through two fixed points, 

 and 0', then the extremities of the base of this triangle trace out 

 the required homographic systems. One extremity corresponds to the 

 given instantaneous screw ; the other is the impulsive screw. The 

 relation is not usually an interchangeable one. If Q be the impulsive 

 screw corresponding to P as an instantaneous screw, then to find the 

 instantaneous screw corresponding to P as an impulsive screw, we 

 must draw PO and PlO', thus determining T, which corresponds to 

 the required instantaneous screw. The line 00' of course inter- 

 sects the circle in points which correspond to the principal screws 

 of inertia. 



It is, however, to be noticed, that if in one case the relation of the 

 screws as instantaneous and impulsive be interchangeable, then it must 

 be interchangeable in every case. In these circumstances, any chord 

 through the pole of the homographic axis intersects the circle in a pair 

 of points so related. 



A somewhat paradoxical case may be glanced at. The polar of 0' 

 cuts the circle in two points, and if either of these points be regarded 

 as coiTesponding to an instantaneous screw, then the impulsive wrench 

 will, fi'om the foregoing construction, lie on a reciprocal screw ; but fi'om 

 the nature of the reciprocal screws it would seem impossible that an 

 impulsive screw and an instantaneous screw could be so related. The 

 paradox is explained by the fact that the instantaneous screw is here 

 imaginary, and possesses the curious property, that even with an unit 

 of twist velocity the kinetic energy is zero. Under these circum- 

 stances, the only way of preventing a finite impulsive wrench from 

 generating an infinite twist velocity is to have the screws reciprocal. 



As the pitch of a screw can be expressed in the fonn^iai^ +p-20-^, 

 and as the kinetic energy for unit twist velocity has the form %iia-^ 

 + u^aJ-, it follows that the law which exhibits the pitch distribution 

 must have a parallel in the law which shows the distribution of kinetic 

 energy. We thus have the result, that a perpendicular fi'om any point 

 of the circle on the polar of 0' is proportional to the kinetic energy due 

 to a unit twist velocity about the corresponding screw. 



In the case of small oscillations with forces having a potential, we 

 find a third point, 0", any chord thi'ough which intersects the circle in 

 points conjugate with regard to the potential. The chord O'O" inter- 

 sects the circle in the two points corresponding with the two harmonic 

 screws. 



