428 Proceedings of the Royal Irish Academy. 



LXIV. PKELnrrNTAET !N'OTE ON THE PlANE EEPEESEIfTATIOlSr OP CeETATN" 



PeOBIEMS IN" THE DxNAMICS OF A ElGID BoDT. Bv EoBEKT 



S. Ball, LL.D., F.R.S. 



[Eead, AprU 11, 1881.] 



The present Paper applies to the case where the rigid body has 

 freedom of the third order, and while the body remains in or near to 

 its original position. Three co-ordinates will then specify any posi- 

 tion which the body can attain. Two independent co-ordinates will 

 specify the screw about which the body is twisting. The screws in 

 the system can be most conveniently designated by three homogeneous 

 co-ordinates of which only the ratios are concerned. The representa- 

 tion of a screw is therefore analogous to the representation of a point 

 in a plane by trilinear co-ordinates. The object of this Paper is to 

 develop this analogy. The reader is presumed to be acquainted with 

 the elements of the Theory of Screios. 



Let 6i, 6n, 62, be the co-ordinates of a screw 6, referred for con- 

 venience to the three principal screws of the three-system, which 

 defines the freedom of the body (see Theory of Screws, p. 116). 

 "We can also denote the position of a point in a plane by the three 

 co-ordinates Oi, 62, O3, and hence we are led to the result that 



To each screio of a three-system corresfonds one point in the plane. 



The converse of this is also yenerally true. It would be universally 

 true but for one conic, and four points thereon, of a very remarkable 

 character. To each of these points corresponds a whole plane of 

 screws in the three-system, while the remaining points on the conic 

 have no screws corresponding to them. 



Two screws determine a cylinclroid, and the co-ordinates of any 

 third screw on the cylindroid are linear functions of the co-ordinates 

 of the two given screws ; and hence 



To each cylindroid of screivs lelonyiny to the three-system corresponds 

 a straiyht line in the plane. 



It is well known that any two cylindroids of a three-system have 

 one common screw. This theorem becomes sufficiently obvious when 

 the cylindroids are represented by right lines, the common screw of 

 course corresponding to their intersection. 



If Pa, p^, Py, be the pitches of the three principal screws of the 

 system {Theory of Screivs, p. 121), then the pitch pe of any other 

 screw 9 is given by the equation 



PaOi'' +P^e,'+Pye,' -Pe (^r + 0,' + ei) = o. 



If we regard pe as given, then this equation corresponds to a conic 

 section in the plane. To each pitch p>e will correspond a different 



