VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 11 



or system of invariable form*, and the investigation of its motion about the centre of 

 gravity requires only the determination of its axis of rotation and the intensity of rotation 

 about that axis. 



If the arrangement be variable, the laws of its variation must be given, and according 

 to the number of possible laws will be the number of different solutions of the problem : 

 here then the problem diverges into special problems; such as that of the motion of a body 

 expanding or contracting according to a given law and the like, where the law of variation 

 is geometrically expressed ; and such as the problems of the motion of fluids, of elastic bodies, 

 or of systems of bodies like the solar system, where the law of variation is mechanically 

 expressed by defining the nature of the internal actions and reactions of the system. We 

 shall confine our attention to the simpler problem of the motion of a system of invariable 

 form, which we proceed to discuss. 



21. The motion of an invariable system is always reducible to the motion of translation 

 of some point invariably connected with it combined with a motion of rotation about a certain 

 axis through that point. Let v^, v^, Vz denote the resolved velocities along Ox, Oy, 0% of 

 the point O, to which the translation is referred, and let w^, Wy, w, denote the resolved angular 

 velocities about the same lines ; then the velocity of any particle m, whose co-ordinates are 

 X, y, ss, is v^ + Wy% - w^y in the direction of Oof, with similar expressions for the directions 

 Oy, Osx. Hence summing the linear and angular momenta of the several particles of the 

 system, we find 



u^ = 2(m) .v^ + cD^. '2(mz) - W;,'2,(my), 



' I avoid the use of the terra rigid body because of the 

 mechanical notion conveyed in the term rigid. The pro- 

 positions usually enunciated with reference to a rigid body 

 must, if that term be retained, be understood of & geometrically, ■ 

 not a mechanically, rigid body ; that is, of a body the disposi- 

 tion of whose parts is by hypothesis unaltered, not of one in 

 which the disposition cannot be altered or can only be insensibly 

 altered by force applied to it. But it is difficult (and perhaps 

 not desirable) to divest this term of its mechanical meaning, 

 as is seen in the modes of expression commonly adopted in the 

 case of flexible strings, fluids, &c., where it is frequently de- 

 manded of us to suppose our strings to become inflexible, our 

 fluids to become rigid, or to be enclosed in rigid envelops, and 

 the like — a process which must always stagger a beginner and 

 leave a certain want of confidence in his results, until this is 

 gained by familiarity with the process, or until he learns that it 

 simply amounts to asserting that what has been laid down to 

 be true of a rigid body is no less true of a non-rigid body, 

 while there is no change in the disposition of its parts. As 

 another instance of a needless limitation in our current defini- 

 tions, we may cite that of Statics as the science which treats of 

 the equilibrium of forces, whereas the truer view would be to 

 regard it as treating of those relations of forces which are inde- 

 pendent of time, and thus every dynamical problem would have 

 its statical part in which the state of the system and the forces 

 is considered at each instant, and its truly dynamical part in 

 which the changes effected from instant to instant are deter- 



mined. This view presents Statics as a natural preparation for 

 Dynamics, instead of as a science of coordinate rank separated 

 by a gulf to be bridged over by a fictitious reduction of dy- 

 namical problems to problems of equilibrium through the intro- 

 duction oi fictitious forces. In several of our more recent works 

 the terms accelerating force and centrifugal force have been 

 rejected or explained as mere abbreviations, the one as not 

 being properly & force, the other as being a fictitious and not an 

 actual force : this it would be well to carry out still more com- 

 pletely, to restrict force in fact to that which is expressible by 

 weight and to admit only actual forces (to the exclusion oi cen- 

 trifugal forces, effective forces and the like) under the two 

 divisions of internal forces, or those whose opposite Reactions 

 are included within the system, and external forces, or those 

 whose opposite Reactions axe not so included. If then Statics 

 and Dynamics were defined as above, one great division of 

 Rational Mechanics would be formed of the Statics and Dyna- 

 mics of a system of given invariable form, without the par- 

 ticular constitution of the system being defined and there- 

 fore independent of Internal Forces ; while the other great 

 division would include the Statics and Dynamics of special 

 systems of defined constitutions, as flexible bodies, fluids, 

 elastic solids and the like, in which the laws of the internal 

 forces must be more or less completely known. These re- 

 marks are thrown out as suggestions for a more natural 

 system of grouping the special mechanical sciences than has 

 yet been commonly received. 



2—2 



