32 Proceedings of the Royal Irish Academy. 



what remarkable when viewed in the present method. Let he the 

 pole of the pitch-line LM, with regard to the circle ; then the theorem 

 to he proved is that any chord through intersects the circle in a pair 

 of points which correspond with a pair of reciprocal screws on the 

 cylindroid. It will be nearly as easy to prove this in a more general 

 form, viz., that the virtual coefficient of any two screws is propor- 

 tional to the perpendicular OR, let fall fi'om on the chord PQ, 

 joining the two corresponding points. 



From the figure, it is very easily seen that 



OR = m cos {I - V) - O^cos {I + V) ; 

 whence 



— OR = p cos / cos I' +PP cos m cos m'; 

 m 



but the right-hand member is the virtual coefficient of the two screws 

 (" Screws," p. 37), and hence the required theorem has been proved. 



It is now easy to find the screw on a cylindi'oid reciprocal to a 

 given screw; for, join the point corresponding with the given screw 

 to the pole of the pitch-line, and the point where the chord cuts the 

 circle again is the correspondent of the required screw. If the screws 

 have a given virtual coefficient, then the chord joining them envelops 

 a circle whose centre is the pole of the pitch-line. 



The graphical method of this Paper is also very convenient for the 

 illustration of the dynamical questions of impulsive forces and of small 

 oscillations. If a rigid body with two degrees of freedom be at rest, 

 and if it receive an impulsive wrench on any arbitrary screw, then it 

 is well known that without any sacrifice of generality we may replace 

 the given impulsive wrench by a wrench on a screw of the cylindroid 

 expressing the fi'eedom (" Screws," p. 59). We thus have two corre- 

 sponding systems of screws on the cylindroid, and the correspondence 

 being strictly of the one-to-one type is homographic ("Screws," p. 106). 

 We thus have in the present way of looking at the subject two sets of 

 homographic point-systems on the circle. We shall call Ri, P^, &c., 

 the impulsive screws, and Qi, Q^, &c., the corresponding instantaneous 

 screws. The general theory of such point-systems on a circle is, of 

 course, well known, and it may be of interest to trace how in the pre- 

 sent case the homography departs from the general type. 



If we join Pi, Qn, and also Po, d, the joining lines intersect on a 

 point coUinear with the similar intersections obtained by taking every 

 other two pairs of impulsive and instantaneous screws. The axis thus 

 obtained intersects the circle in the two double points of the homo- 

 graphy. The screws corresponding to these points are what we have 

 called the principal screws of inertia. They are, in fact, the screws, a 

 wrench on either of which will force the body to commence its move- 

 ment by twisting around the same screw. Now, although this axis 

 may remain arbitrary in the case of a perfectly general homography, 

 yet, in the special kind of homography now before us, this axis must 



