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XIII. On the Dynamics of a Rigid Body in Elliptic Space. 
By P. S. Heath, B.A., Fellow of Trinity College, Cambridge. 
Communicated by Professor Cayley, Sadlerian Professor of Mathematics in the 
University of Cambridge. 
Received January 4,—Read January 17, 1884. 
This paper is an attempt to work out the theory of the motion of a rigid body under 
the action of any forces, with the generalised conceptions of distance of the so-called 
non-Euclidean geometry. Of the three kinds of non-Euclidean space, that known as 
elliptic space has been chosen, because of the perfect duality and symmetry which 
exist in this case. The special features of the method employed are the extensive use 
of the symmetrical and homogeneous system of coordinates given by a quadrantal 
tetrahedron, and the use of Professor Cayley’s coordinates, in preference to the 
“rotors” of Professor Clifford, to represent the position of a line in space. 
The first part, §§ 1-21, is introductory; in it the theory of plane and solid geometry 
is briefly worked out from the basis of Professor Cayley’s idea of an absolute quadric. 
By taking a quadrantal triangle ( i.e ., a triangle self-conjugate with regard to the 
absolute conic) as the triangle of reference, the equations to lines, circles, and conics 
are found in a simple form, and some of their properties investigated. 
The geometi’Y of any plane is proved to be the same as that of a sphere of unit 
radius, so that elliptic space is shown to have a uniform positive curvature. 
The theory is then extended to solid geometry, and the most important relations of 
planes and lines to each other are worked out. 
The next part treats of the kinematics of a rigid body. The possibility of the 
existence of a rigid body is shown to be implied by the constant curvature of elliptic 
space, and then the theory of its displacement is made to depend entirely on ortho¬ 
gonal transformation. Any displacement may be expressed as a twist about a certain 
screw. A rotation about a line is shown to he the same as an equal translation along 
its polar; so that the difference between a rotation and a translation disappears, and 
the motion of any body is expressed in terms of six symmetrical angular velocities. 
An angular velocity a>, about a line whose coordinates are a, b, c, f, g, h, is found to 
be capable of resolution into component angular velocities, aco, bu . . . haj, about the 
edges of the fundamental tetrahedron. 
The theory of screws is next considered. A twist on a screw can be replaced by a 
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