220 On Dynamics of a Rigid Body in Elliptic Space. [Jan. 17, 



the extensive use of the symmetrical and homogeneous system of 

 co-ordinates given by a quadrantal tetrahedron, and the use of Pro- 

 fessor Cayley's co-ordinates, 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 conies are 

 found in a simple form, and some of their properties investigated. 



The geometry 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 possi- 

 bility 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 displace- 

 ment is made to depend entirely on orthogonal transformation. Any 

 displacement may be expressed as a twist about a certain screw. A 

 rotation about a line is shown to be the same as an equal translation 

 along its polar ; so that the difference between a rotation and a trans- 

 lation disappears, and the motion of any body is expressed in terms of 

 six symmetrical angular velocities. An angular velocity w, about a line 

 whose co-ordinates are a, b, c, f, g, h, is found to be capable of resolu- 

 tion into component angular velocities aoo, biv . . . htv, about the edges 

 of the fundamental tetrahedron. 



The theory of screws is next considered. A twist on a screw can be 

 replaced by a pair of rotations about any two lines which are conju- 

 gate to each other in a certain linear complex. The surface corre- 

 sponding to the cylindroid is found to be of the fourth order with 

 a pair of nodal lines. Lastly, the condition of equivalence of any 

 number of twists about given screws is investigated. 



In kinetics, the measure of force is deduced from Newton's second 

 law of motion, and the laws of combination and resolution are proved. 

 The consideration of the whole momentum of a body suggests the 

 idea of moments of inertia, and a few of their properties are investi- 

 gated. The general equations of motion referred to any moving axes 

 are then found, and in a particular case they reduce to a form corre- 

 sponding to Euler's equations ; these are of the type 



A4- (B-H)^ 6 -(a-C)a^ 3 =Q 1 . 



The last part is occupied in the solution of these equations when no 

 forces act, in terms of the Theta-functions of two variables. A solu- 

 tion is obtained in the form 



