434 Proceedings of the Royal Irish Academy. 



It follows at once {Theory of Screivs, p. 136) that we must have 



where -2" is proportional to the kinetic energy. The required locus is 

 therefore a conic having double contact with the inertia conic. 

 It is easy to prove from this that S will be a maximum if 



U^6l '-PaVl = 'th% ••PpV2 = ii3% '• Py'm\ 



whence we have Euler's well-known theorem that if the bodybe allowed 

 to select the screw about which it will twist, the kinetic energy acquired 

 will be larger than when the body is constrained to a screw other than 

 that which it naturally chooses. 



A somewhat curious result arises when we seek the interpretation 

 of a tangent to the inj&nite pitch conic. This tangent must, like any 

 other straight line, correspond to a cylindroid; and since it is the polar 

 of the point of contact, it follows that every screw on the cylindroid 

 must be at right angles to the direction corresponding to the point of 

 contact. The co-ordinates of the point of contact must therefore be 

 proportional to the direction cosines of the nodal line of the cylindroid. 



If the body be in equilibrium under the action of a conservative 

 system of forces, then there is a conic (analogous to the conic of iner- 

 tia) which denotes the locus of screws about which the body can be 

 displaced to a neighbouring position, so that even as far as the second 

 order of small quantities no energy is consumed. The vertices of the 

 common self conjugate triangle of this conic and the conic of inertia 

 correspond to the harmonic screws about which, if the body be once 

 displaced, it will continue for ever to oscillate. 



The further development of the subject, on which this Paper is a 

 preliminary note, must form the basis of a future and more extensive 

 memoir. 



