Ball — On Problems in the Dynamics of a Rigid Body. 431 



The vertices of the self-conjugate triangle that can be constructed 

 with respect to the two conies correspond to screws mutually recipro- 

 cal and rectangular, whence 



The principal screivs of the three-system correspond to the vertices of 

 the triangle ivhich is self -conjugate ivith regard to the zero-pitch, and the 

 infinite-pitch conic, 



For the determination of a three-system nine data are required — 

 for example, nine data will give the pitch quadric, and when that is 

 known the rest of the system is determined. An equal number of 

 data is required for the plane representation. Pive of these may con- 

 veniently specify the zero-pitch conic, while the specification of the 

 four fundamental points thereon will absorb the remainder. 



The conic and four points being known, the self-conjugate triangle 

 is determined ; the equation of the conic referred to that triangle is 

 therefore known, and thus the pitches pa., p^, Py of the three princi- 

 pal screws are determined. It remains to be shown how the pitch of 

 the screw corresponding to any other point in the plane is to be ascer- 

 tained. 



It is not difficult to prove the following theorem : — 



Measure ojf distances p>a, P^, Py, Pe on a straight line from an arli- 

 tra/ry point, then the anharmonic ratio of the four points thus obtained is 

 equcd to the anharmonic ratio tohich the point corresponding to 6 sultends 

 at the four fundamental points. 



"We are now able to construct the infinite-pitch, or any other pitch 

 conic, from the primitive data, as the problem is merely to draw a 

 conic through four points so that the anharmonic ratio subtended at 

 those four points by a variable point shall be given. 



Each screw of a three-system has one screw of the reciprocal sys- 

 tem parallel to it, with a pitch of changed sign. If we take a plane 

 representation and change the signs of all the pitches, then the new 

 arrangement gives the screws of the reciprocal system parallel to all 

 those of the old. 



Two conies can be described through the four fundamental points 

 to touch any given straight line; the two points of contact will 

 indicate the two principal screws on the cylindroid corresponding to 

 the straight Kne. The other pitch conies will cut the line in points 

 which form an involution, each pair corresponding to the two screws 

 of the same pitch on the cylindroid. 



The polar of a point with regard to the zero-pitch conies corre- 

 sponds to the cylindroid which is the locus of screws in the three-system 

 reciprocal to a given screw. 



On each cylindroid one screw <^ reciprocal to a given screw 6 can 

 be found. It is only necessary to take the polar of 9 with regard to 

 the zero-pitch conic, and the point in which it intersects the line cor- 

 responding to the cylindroid gives the required screw. 



By the aid of the plane representation we are enabled to solve many 



