I30 



NA TURE 



[June 5, 1890 



points in a plane. This has been worked out in chapters 

 xxi. and xxii. In fact, here the screw co-ordinates, three 

 in number, are simply taken astri-linear co-ordinates of a 

 point. It follows at once that the locus of points with 

 equal pitch must be a conic, the " absolute ' being the 

 locus of pitch 00, and one conic relates to zero-pitch. 

 This latter may, without loss of generality, be made a 

 circle. 



It is of interest to notice that for a screw-complex of 

 order 3 the screws which have a given pitch form them- 

 selves a quadric surface, viz. they form one set of gene- 

 rators on a hyperboloid, the other set of generators 

 having pitch - p, and containing thus screws in the com- 

 plex reciprocal to the others. 



Other quadrics enter the theory, especially one con- 

 taining the locus of screws about which a body may twist 

 without receiving kinetic energy, and which is, of course, 

 imaginary; and one connected with the potential. These 

 last two determine the principal screws of inertia, of which 

 more later on. 



For screw-complexes of order 4 no graphical repre- 

 sentation is given. The difficulty lies here in this — that 

 the dynamics require constantly metrical relations, and 

 these are not very simple in the plane representation, by 

 conies for instance. It is here that the non-Euclidian 

 character of the geometry comes out. 



These speculations are in close connection with the 

 contents of chapter xix., where projective relations between 

 two congruences of the same order are investigate'd. It is 

 here that Herr Gravelius has more particularly introduced 

 original work of his own in bringing Sir R. Ball's Mr. 

 One-to-one more prominently to the foreground. 



Up to this we have considered chiefly the geometry 

 of systems of screws. It is now time to consider the 

 kineniatics of a rigid body and the action of forces on it. 



If a body is perfectly free it can twist about every 

 screw in space. As these can be decomposed into six 

 twists about the co-ordinate screws, the body is said to 

 have six degrees of freedom. If the body is constrained 

 in any manner — and here the generality of the nature of 

 the constraint has to be noticed —then it will not be able 

 any longer to twist about all screws. But we have seen 

 already if it can twist about n screws it can twist about 

 all screws belonging to the complex of order n derived 

 from them. The freedom of a body is therefore fully 

 characterized by the complex which contains all screws 

 about which the body can twist. If this is of order n, 

 then the body has n degrees of freedom. An attempt to 

 twist the body about any other screw will evoke a re- 

 action due to the constraint which will reduce to a wrench 

 upon some screw. Such a wrench cannot do any work 

 against a possible twist of the body, hence the screws 

 on which wrenches of constraint are possible must be 

 reciprocal to the screws which determine the freedom of 

 the body ; they form, therefore, the reciprocal complex. 

 We thus get the very general theorem about the equi- 

 librium of a body. If a body has n degrees of freedom 

 then it will be in equilibrium under the action of all 

 wrenches on screws of a certain complex of order 6 - n. 

 This complex may be called the complex of constraint. 



Again, if a body is subjected to an impulsive wrench 

 'upon a screw, >;, not belonging to the complex of constraint, 

 it will begin to turn about some screw, a, called the instan- 

 taneous screw. At the same time an impulsive wrench of 

 constraint will be evoked. Conversely, in order to produce 

 a twist on u as instantaneous screw we may apply an 

 impulsive wrench on 77, but with this we may combine a 

 wrench on any one of the screws belonging to the complex 

 of constraint. As the latter is of order 6 — n, all screws 

 derivable from these, together with the screw rj will form 

 a complex of order j — n This complex of order j — n 

 and the complex of order 11 which determine the freedom 

 have y-n-\-n-6=i screw in common (see above). 

 This screw is called the reduced impulsive wrench. 



NO. 1075, VOL. 42] 



We thus have proved if a body has freedom of order ?t, 

 then there is always one and only one screw, jj, in the com- 

 plex which determines the freedom, such that an impulsive 

 wrench on it makes any given screw, a, the instantaneous 

 screw. The converse, also, is evidently true. Between the 

 impulsive and instantaneous screw in the complex exists, 

 therefore, a one-one correspondence, or, to express this 

 differently, the complex of instantaneous and that of im- 

 pulsive screws are projective. They are also coincident. 

 But if we have two coincident projective spaces of « — i 

 dimensions, then there are always n screws in one which 

 coincide with their correspondents. This proves if a 

 body has n degrees of freedom, then there exist n 

 screws, and in general only n, such that an impulsive 

 wrench on one of them produces a twist on the same 

 screw. These n screws— and the discovery is one of 

 the triumphs of the theory — are called the principal 

 screws of inertia, as they depend on the distribution of 

 matter in the body. These screws are also co-reciprocal, 

 and may therefore be taken as co-ordinate screws. They 

 are a generalization of the principal axes of inertia in the 

 ordinary theory ; and to show their importance it is suffi- 

 cient to point to the importance of the principal axes of 

 inertia in the ordinary theory of a free body, or of a body 

 of which one point is fixed, and to remember the simpH- 

 fication obtained by taking them as axes of reference. 



For a free body the screws of inertia lie on the prin- 

 cipal axes of the body which pass through the mass- 

 centre, two on each, with pitches equal to the correspond- 

 ing radius of gyration, taken positive for the one and 

 negative for the other. The ordinary theory has no 

 analogon to this if the body is constrained, excepting in 

 the few cases where a point or an axis of the body is 

 fixed, or where the body has plane motion only. 



It is in such generalizations that the theory of screws 

 excels. It has given us here the best and simplest co- 

 ordinates for all cases of the motion of a single rigid 

 body acted on by any forces and constrained in any 

 manner conceivable. 



We will now suppose that the co-ordinates thus pointed 

 out are used, and find the instantaneous screw corre- 

 sponding to any given impulsive wrench. Each com- 

 ponent wrench produces a twist about its own screw, whose 

 amplitude depends in a very simple manner on the in- 

 tensity of the impulsive wrench ; so that the intensities 

 of the component twists are known, and these give the 

 resultant twist. 



We next consider the kinetic energy, T, of the body 

 due to a twist on a screw, a. Let aj, o,,, ... be its com- 

 ponents, /j, />2i • • • the pitches of the co-ordinate screws, 

 and d the twist velocity. It is then shown that, M being 

 the mass of the body. 



T = Ma-(A''«l' + /2'«2 



-f p,ran 



Denoting the expression in the brackets by Ua^, we 

 have T = Ma"ua-. The quantity tia is a length ; the 

 expression for T is therefore of the same form as that for 

 the rotation of a body about an axis with angular velocity 

 «, the radius of gyration being replaced by Uaj >j'2. This 

 last expression deserves a name. If we adopt Clifford's 

 word " spin-radius," instead of radius of gyration, the 

 name twist-radius suggests itself as suitable for Ua or 



We now come to consider the problem of small oscilla- 

 tions. Let there then be a body of n degrees of freedom 

 in a position of equilibrium under a system of forces which 

 have a potential V. Let A denote the complex defining 

 the freedom. If the body be displaced by a small twist 

 about a screw, a, belonging to the complex A, then the 

 forces are not any longer in equilibrium ; hence they will 

 give rise to a wrench on some screw A. This wrench 

 may be combined with any wrench of constraint ; but just 

 as in case of impulsive wrenches there is one single screw 



