Ball — On Prohlems in the Dynamics of a Rigid Body. 429 



conic. All the conies will form a family of the type 8 + kS' = 0, and 

 they intersect in the same four points. These points are defined by 

 the equations 



Oi' + ^o' + O-i" = 0. 



The first of these equations denotes the conic -vrhich corresponds to 

 the screws of zero pitch. The second of the equations denotes the 

 locus of the screws of infinite pitch. It is the exceptional conic just 

 referred to. The four points common to their conies are of indeter- 

 minate pitch; they are indeed the exceptional points, and of course 

 they are imaginary. 



It has been shown (Theory of Screws, p. 121) that the locus of the 

 screws of pitch 2^e in the three-system is the quadric 



(Pa -Pe ) ^^ + (P^ - Pe) y- + (Py -Pe) s^ + (Pa -Pe)(pp - Pe){Py - Pb) = 0. 



This is the real part of the locus, but the complete locus contains an 

 imaginary portion also. This fact is at once exhibited by the plane 

 representation. A straight line in the plane represents a cylindroid, 

 «'. e. a surface of the third degree. It hence followed that a conic in 

 the plane should correspond to a surface of the sixth degree. We 

 thus learn that the real locus of the screws of any given pitch must 

 be a surface of the sixth deyree, and that consequently the quadric with 

 which we were already acquainted requires to be multiplied by a fac- 

 tor of the fourth degree. 



It can be shown that this factor is the product of the four planes^ 

 produced by giving variety of sign to the coefficients in 



"/p^ -PyX \ V p^ - Pa y + "/ Pa -p^r. 



+ V" Pfi -Py'y Py -Pa V Pa. - P^ = 0. 



In general, ii x, y, z be the co-ordinates of a point on a screw 

 belonging to the system, and if di, 60, O3 be its co-ordinates, then we 

 have 



^(^,2 + ^3') -ye,e,-ze,es + {p^ - py) e, e. = o, 

 y(^3^ + 6^-) -ze,6,-xe,e, + {p^ -pa)e,e, = o, 

 s (^1- + 0.^) -xe,e,-y6,6, + {pa - Pi) ej.^o. 



In the present case 



61" : 0.' : 62' :: p^ - p^ : p^ - p„. : Pa - p^, 



