46 Proceedings of the Royal Irish Academy. 



The plane of the resultant couple will be perpendicular to the resultant 

 force if ^L ^ VXp = p\, (44) 



where p is some scalar. In this case the moment of the resulting couple 

 is pTX. 



The intensity of the resultant force is TX ; hence p is the ratio of the 

 moment of the couple to the intensity of the force ; i.e. p is the pitch of the 

 screw on wliich the given system of forces forms a wrench. 



From the equation (44) we have 



^\-' + T^p.X-' = /), 

 whence taking scalai-s ]} = .S'/iX"'; (45) 



and we obtain the instructive result thus stated. 



If (ji, A) be the resultant couple and resultant force of any system of 

 forces applied to a rigid body, then the resultant wrench is on a screw of 

 which the pitch is S/iX'K 



We can now expi-ess the vector-perpendicular from on the screw in 

 question. If /] be this vector, then the equation (44) may be written 

 n + V\p 4 S\p = p\, because SXp = 0. 



We thus have 



/« + Xp " pX, or X-'n + p ^ p, or p ^ - VX'^fi = Vfi\''; (46) 



we thus have another result also of the greatest importance in our present 

 subject, which may be thus stated : — 



If (/i, A) be the resultant couple and resultant force of any system of 

 forces applied to a rigid body and with respect to any point, then the vector 

 from the origin perpendicular to the screw on which the system of forces 

 forms a wrench is expressed by V/iX'K This result, as well aa the corre- 

 sponding value of the pitch, (45) is due to Joly,* though they are essentially 

 deductions from Hamilton's quaternion expression for a system of forces.f 



The coordinates (ji, A) define not merely a screw, they define a vector- 

 screw ; for UX will indicate which of the two vector-screws on the same axis 

 is to be understood ; and TX expresses the intensity of the dyname of which 

 the vector-screw is the site. Thus the completeness of the quaternion re- 

 presentation of the dyname by the two coordinates (ji,X) leaves nothing more 

 to be desired. 



We can now obtain the quaternion equation of the screw on which the 

 dyname with coordinates (/*, A) is situated. As the screw is parallel to A 

 and as F^iiA"' is a point on the screw, we must have for the vector p to any 

 point on the screw p = Vp-X'^ + t\ (47) 



where < is a variable scalar. 



• " Manual," p. 156. t "Elements," toI. ii., p. 285. 



