January io, tgoi] 



NA TURE 



247 



of freedom can only move on one definite screw. A body 

 with two degrees of freedom can take two independent 

 screw motions combined in any ratio. This gives an 

 infinite number of resultant screws, all lying in one ruled 

 surface (called a cylindroid) and having pitches distri- 

 buted according to a simple law. All cylindroids have 

 the same shape, and the linear dimensions of a cylindroid 

 are proportional to the difference between the greatest and 

 least pitch that can be found among its screws. A body 

 with }i degrees of freedom has n independent screw 

 motions, n being not greater than 6. 



Any screw system which specifies the freedom of a 

 body serves equally well for specifying aggregates of 

 screw-actors of the forcive kind. For instance, if any 

 multiple (integral or fractional) of a forcive on a screw A 

 is compounded (additively or subtractively) with a forcive 

 on a screw B, the resulting forcive will be on one of the 

 screws of the cylindroid to which A and B belong. 



A body limited to motion on one definite screw can 

 move in either of two opposite directions, and when acted 

 upon by a forcive will move in the direction for which 

 the work done by the forcive is positive. When the 

 forcive-screw is so related to the motion-screw that the 

 work-rate for a small motion is zero, the body will be in 

 equilibrium, the forcive being equilibrated by the reaction 

 of the constraints. The two screws in this case are said 

 to be reciprocal. The condition of reciprocity, when ex- 

 pressed in terms of the rectangular components X Y Z of 

 force, uv w oi translation, L M N of couple, and i> q r 

 of rotation, is 



X« -f Yz/ -1- Zw-h L/ -I- M$r -I- Nr= o ; 



and this will not be altered by interchanging the force X 

 with the rotation/, the couple L with the translation u, 

 and so on. Hence it is immaterial which of the two 

 screws we assign to the forcive and which to the motion. 



One degree of constraint subjects the 6 components, 

 u V w p g r, oi axi instantaneous motion to one linear re- 

 lation. It can accordingly be expressed by assigning 

 proper values to the coefficients X Y Z L M N in the 

 above equation of condition. One degree of constraint, 

 therefore, limits the motion of a body to those screws 

 which are reciprocal to one definite screw. Hence it can 

 be deduced that two degrees of constraint limit the 

 motion to screws reciprocal to all the screws of one de- 

 finite cylindroid. 



The system of screws on which a body can move which 

 has « degrees of freedom or 6-« degrees of constraint 

 being called an n system, it can be shown that every n 

 system is reciprocal to a definite 6 -« system. Each of 

 these two systems is sufficient to define the other. 



In a 5 system the axes of the screws include every line 

 in space, each fitted with its proper pitch, and at every 

 point there are a whole plane of screws of any assigned 

 pitch. 



Three or more screws are said to be independent when 

 it is not possible to take screw-actors upon them whose 

 sum is zero. Seven screws cannot be independent. If 

 any 6 independent screws are taken, they will suffice for 

 the specification of any 7th screw by 6 numerical co- 

 efficients, called screw-coordinates, an actor on the 7th 

 screw being always a sum of multiples (generally frac- 

 tional) of actors on the 6 screws of reference. It is pos- 

 NO. 1628, VOL. 63] 



sible, and usually preferable, to select screws of reference 

 such that each is reciprocal to all the rest. 



If a body is only free to move on a single screw, a 

 forcive applied to it can be resolved into two forcives, one 

 of which is reciprocal to the screw in question. This 

 component can be ignored, as it does not influence the 

 motion, which will accordingly be the same as if the 

 other screw acted alone. If a body has n degrees of free- 

 dom, a forcive applied to it can be resolved into 6 mutu- 

 ally reciprocal forcives of which 6-« are without influence 

 on the motion, and the other n may be regarded as acting 

 alone. 



The initial motion of a body produced by an impulsive 

 forcive is, in general, on a different screw from the forcive ; 

 but in certain cases they are on the same screw. (This 

 means that they have the same axis, and the work in 

 translation is equal to the work in rotation). The screw, 

 common to both, is then called a principal screw of 

 inertia. There are, in general, 6 such screws for a per- 

 fectly free body, and n for a body with n degrees of 

 freedom. 



Again, for a body in stable equilibrium under forces 

 which have a potential, there are certain screws (gener- 

 ally equal in number to the degrees of freedom) such 

 that if the body be slightly displaced along one of these 

 screws, and then left to itself either at rest or with a 

 velocity on the same screw, it will oscillate on this screw. 

 The screws thus defined are called harmonic, and are the 

 proper screws to select for specifying small oscillations. 



Besides physical deductions, of which the foregoing 

 are specimens, the treatise contains numerous geometri- 

 cal investigations, and an extension of the theory to non- 

 Euclidian space. 



At the end of the volume an interesting summary is 

 given of the literature of the subject. It appears that 

 Hamilton, in one of his papers on systems of rays, 

 and Pliicker, in his New Geometry of Space, anticipated 

 Sir R. Ball's discovery of the cylindroid so far as regards 

 its geometrical form without reference to pitch ; and 

 several theorems respecting systems of lines had been 

 discovered which are particular cases of the general 

 theory of screws. 



An amusing and instructive " Dynamical Parable," 

 which formed Sir R. Ball's Presidential Address to 

 Section A at the 1887 meeting of the British Association, 

 is given as an Appendix. 



I wish to point out an erroneous statement with regard 

 to finite displacements which occurs in all our works of 

 highest authority on the motion of a rigid body. It is to 

 be found in Routh, in Thomson and Tait, in Williamson 

 and Tarleton, and in the introduction to Sir R. Ball's 

 Treatise. The erroneous statement, in its plainest shape, 

 is "The same displacement cannot be constructed on 

 two different screws." 



To see that this is wrong, consider the effect of giving 

 a nut 9j turns on an ordinary iron screw. The same 

 final position could obviously be attained by employing 

 a screw of longer pitch and taking fewer turns, say Z\ or 

 \\ or \, and could also be attained by taking \ or i| or 

 8| turns on a left-handed screw. The correct statement 

 is that the axis and translation are unique, but that the 

 rotation has any one of an indefinite number of values 

 differing each from the next by 27r, some of them being 



