(54 Proceedings of the Royal Irish Academy. 



the reader is acquainted with the principal properties of linear vector 

 functions and also with the geometrical properties of the 3-system.* 



"We shall first prove the following general proposition : — 



If <^ be a linear vector function, then 



p ^ \{V,^\ .\-^ - V<^'\ . X-') + .rX 

 is the equation to a diameter of the pitch-quadric of the 3-system defined 

 by the function ^. 



A screw q parallel to A can of course be found in the 3-system S and 

 its pitch p is 5^X .X"'. A screw % parallel to X can also be found in the 

 reciprocal 3-system ^ , and its pitch is 



- iyX . X' = - S^\ . X'. 

 Hence the pitches of the screws parallel to X in the S-system and its 

 reciprocal 3-system differ merely in sign. These two screws are therefore 

 parallel generators of the hyperboloid which expresses the locus of the screws 

 of pitch /> contained in the system S. A parallel to these generators drawn 

 through the point midway between them must therefore be a diameter of 

 the />-pitch-quadric, and therefore of the zero pitch-quadric, for, whatever p 

 may be, the ^-pitch^uadric is concentric with the zero pitch-quadric. 



O 



Pio. 16. 



Let the plane of the paper pass through the origin and be perpendicular 

 to the two screws ij and £ which meet the plane in ^, ^ respectively 

 (fig. 16). The point C bisects AB, and the line through C perpendicular to 

 the plane of the paper is therefore a diameter of the zero pitch-quadric. As 

 OA is the perpendicular from the origin on 17, we must have the vector OA 

 equal to F^X . X''. In like manner OB Sa - F^'X . X"' ; and by completing 

 the parallelogram, we have for the vector OC 



OC = \{V^\.\-' - FfX.X-), 

 and this ia the vector from the origin perpendicular upon that diameter of the 



See •• Trwuise." pp. 170-194. 



