28 REPORT — 1873. 



a screw a be decomposed into six components, ec^, Sec, a^, about the coreciprocals, 

 then «;, &c., Og, are the coordinates of a. 

 The pitch of a is 



where py, &ii.,Pg, are the pitches of the coreciprocals. 

 The condition that two screws a, ^ are reciprocal is 



2. Impulsive and instantaneous Screws. — By proper selection of the coreciprocal 

 group the relation between an impulsive screw and the coiTesponding instanta- 

 neoiis screw is very simple. If «[, &c., a^, be an instantaneous screw, thenp,«,, 

 &c., Pf^a^, is the correspouding impulsive screw. Two of the coreciprocals are 

 directed along each of the principal axes through the centre of inertia of the rigid 

 body ; and the corresponding pitches are + and — the radius of gyration. 



3. Conjugate Screws of Kinetic energy.— \i 



then the impulsive screw corresponding to a is reciprocal to ft ; but precisely the 

 same condition expresses that the impulsive screw con-esponding to |3 is recipro- 

 cal to a. 



On the Kinematics of a Rigid Body*. By Professor J. D. Everett, F.R.S.E. 



The object of the paper is the investigation of the instantaneous movement of a 

 rigid body (having no point fixed). Such investigation has usually been confined 

 to properties depending on the consideration of two consecutive positions ; and the 

 investigation is here extended to properties depending on three, and in the case of 

 motion in one plane to four and five consecutive positions. 



The most general motion of a rigid body may, as is well known, be represented 

 by a succession of small screwings about successive lines called central axes ; and 

 these successive central axes generate two ruled surfaces — one in the body, and the 

 other in space — these two surfaces being perfectly determinate in the case of any 

 given motion. 



Two cones of determinate shape can be constructed by drawing through an arbi- 

 trary point of the body lines parallel to the successive central axes in the body, 

 and by drawing through an arbitrary point of space lines parallel to the successive 

 central axes in space. It is shown in this paper that the most general motion of a 

 rigid body can be represented by giving to the cone in space a motion of pure 

 translation, and causing the cone in the body to roll upon the cone thus translated. 



Expressions are obtained for the curvatures of the two cones coiTesponding to a 

 given instantaneous motion, the data being derived from the consideration of four 

 consecutive positions of the body. ^Yhen only three consecutive positions are 

 given, the curvatures of the two cones are indeterminate, being merely connected 

 by one equation of condition. Hence, so far as regards properties depending on 

 three consecutive positions, the instantaneous motion of a rigid body can always 

 be represented by the rolling of a right circidar cone in the body upon a plane 

 which has a movement of translation in space. In this representation the curva- 

 ture of the circular cone is determinate, but its vertex is an arbitrary particle ot 

 the bod}'. 



The locus of those particles which at the instant considered have straight 

 motion, is investigated, and is found to be in general a cubic curve. 



The cuiwatures of the two ruled surfaces at points on their respective lines of 

 striction are investigated ; and it is shown that the tangent plane to either of the 

 ruled surfaces at a point on the line of striction is perpendicular to the correspond- 

 ing tangent plane of the cone. The forms of the two ruled surfaces, at points very 



* The paper will appear in full in the • Quar(cr!_v Jmimal of Mathematics' for 1874. 



