June 5, 1890] 



NATURE 



131 



X belonging to the complex A, hence now also we have 

 in the complex A a one-one correspondence between the 

 screws a and the screws X. There are therefore, again, 

 n screws a, which coincide with their corresponding 

 screws A. These have got the name of " principal screws 

 of potential." They depend on the system of forces or on 

 the potential V, just as the screws of inertia depend on the 

 distribution of matter. These n screws, again, are co- 

 reciprocal. They have the property that a twist about 

 one of them evokes a wrench on the same screw, the 

 wrench being due to the applied forces. To show the 

 importance of these principal screws of potential it will 

 be sufficient to remark that the potential is, under the 

 circumstance explained, a homogeneous function of the 

 second degree of the n co-ordinates by which the dis- 

 placement is defined. This function becomes the sum of 

 // terms containing the squares only of the co-ordinates 

 if the principal screws of potential are taken as co-ordinate 

 screws. 



Now, suppose that the body has been displaced by a 

 twist about a screw a, this could be done by a wrench upon 

 the screw »;, which as impulsive screw corresponds to a as 

 instantaneous screw. At the same time this displacement 

 calls a wrench on a screw X into play due to the potential 

 \, To every screw a corresponds thus one screw r] and 

 one screw X. Hence the latter are also connected by a 

 1^ one-one correspondence, and there are therefore n screws 



a such that the corresponding screws j; and X coincide. 

 The screws a thus obtained are called '' harmonic screws." 

 They possess this property : A twist about a harmonic 

 screw evokes a wrench which in its turn tends to produce 

 a twist on the original harmonic screw. Hence if the 

 equilibrium is stable this wrench will tend to twist the 

 body back to the position of equilibrium, and thus pro- 

 duce small oscillations about the harmonic screw. From 

 this we get the following theorem, distinguished again by 

 its great generality : — 



If a rigid body having n degrees of freedom is in a 

 position of stable equilibrium under the action of a system 

 of conservative forces, then it can, on being disturbed, 

 perform n distinct oscillations, which consist each of a 

 twisting about a single screw. Every other oscillation is 

 a combination of these. 



These are the chief results which so far have been 

 obtained by the theory of screws as applied to a single 

 rigid body. They form the contents of chapters vi. to 

 xii. These general results are, in the next six chapters, 

 applied and considered more in detail for each of the six 

 possible cases of degrees of freedom which a rigid body 

 may have. Then there follow four chapters on graphical 

 methods, already referred to. 



All the former investigations relate to one single rigid 

 body. But Sir R. Ball, in i88i, published a paper in 

 which he extends his theory to systems of rigid bodies by 

 a method as beautiful as it is suitable to the purpose. 



The bodies, of which we suppose there are /x, are taken 

 in a definite order. Every body of the system will at 

 every moment twist about some screw. We thus get a 

 et of /x screws, about which at any moment the bodies 

 twist. If we take two consecutive twists, then their '•■ 

 resultant depends only on the ratios of the two amplitudes, 

 and conversely the screw of the resultant determines this 

 ratio. If the screws about which two consecutive bodies 

 twist are given, and also the screw on which their re- } 

 sultant lies, then the amplitude of the first twist deter- 

 mines that of the other. If, therefore, the screws about 

 which the \i. bodies twist at any moment are given, and 

 besides the /x - i screws on which the resultant twists of 

 consecutive bodies lie, then the amplitude of the first 

 determines that of every other twist. The set of 2/1 - i 

 screws thus obtained is called a screw-chain, and it 

 is said that the system of bodies twists at any moment 

 about a certain screw-chain. 



In case of systems of rigid bodies, the screw-chain has 



NO. 1075, VOL. 42] 



to be considered as the fundamental entity, which takes 

 the place of a screw in case of a single body. 



In a finite number of bodies we get a screw-chain of a 

 finite number of screws. These will, in the screw-space 

 of five dimensions, be represented by a finite group of 

 points (elements). If, however, the number of bodies 

 increases and becomes infinite, as in the case of the 

 molecules of a fluid, this group of points may form a 

 continuous locus of one or more dimensions. We may 

 thus get, instead of screw-chains, continuous curves and 

 surfaces of screws, and their geometry will be that of a 

 group of points in five-dimensional non-Euclidian space. 



This suggests an enormous field for investigation, and 

 it is of interest to see that every progress in the algebra 

 and geometry of such a space must indicate also progress 

 in dynamics. 



But these are speculations far beyond the contents of 

 the book under review. 



All results obtained for twists can at once be transferred 

 to wrenches. Accordingly a system of forces acting on a 

 system of bodies can be reduced to a wrench upon a 

 screw-chain. 



There are reciprocal screw-chains, screw-chains of 

 inertia, complexes of screw-chains, complexes of freedom 

 and of constraint, and complexes reciprocal to them. In 

 fact, the screw-chain seems now to take in every respect 

 the place of the screw in the theory of a single body. 

 These screw-chains in their kinematical and dynamical 

 applications to systems of rigid bodies form the contents of 

 the chapters xxiii. and xxiv. 



The last two chapters in the book give Sir R. Ball's 

 theory of content, in which the author tries successfully 

 to overcome the difficulty which ofTers itself in the de- 

 termination of metrical relations without any reference to 

 measuring a length. By " content " is understood the 

 aggregate of all elements in what Clifford called a three- 

 way spread. The investigation is carried on quite alge- 

 braically by aid of the methods of Grassmann's " Aus- 

 dehnungslehre." In the book before us this is worked out, 

 partly with reference to Clifford's theory of biquaternions, 

 and ends with the introduction of Clifford's vectors in 

 non-Euclidian space. 



It will be asked what progress in the science of dyna- 

 mics, and through dynamics in natural philosophy, has 

 been made by Ball's creation. The theory of screws is 

 a mathematical speculation full of life, full of interest and 

 charm for the mathematician who likes to find new 

 physical interpretations for geometrical and algebraical 

 results and methods. The physicist, however, may say 

 that the theory does not increase our power over Nature. 

 But I am inclined to think that when further developed it 

 will be a great, perhaps a very great, help to progress. 

 Does not every molecule of a fluid having rotational 

 motion twist about some screw ? And does not a vortex- 

 line suggest a screw-chain containing an infinite number 

 of elements ? 



The theory of screw-chains, containing a finite number 

 of elements belonging to a system of bodies with one 

 degree of freedom, seems to indicate a truly scientific 

 classification of mechanisms, and may conceivably render 

 great aid in the invention of mechanisms which answer 

 a given purpose. 



The essentially geometrical character of the new method 

 seems particularly well adapted to give graphical solu- 

 tions of dynamical problems, and thus a " graphical 

 dynamics " appears to find here a sound foundation. In 

 this direction much has been done already, but much re- 

 mains to be done. Also the restrictions of infinitely small 

 ampHtudes of the twists has to be broken through, and 

 the infinitesimal calculus has to be pressed into the 

 service. 



Meanwhile, we congratulate Sir Robert Ball on the 

 results which his persevering labours have achieved, and 

 Herr Gravelius on the courage which led him to under- 



