April 13, 1876] 



NATURE 



463 



with respect to the nature of comets and the origin of 

 their light. He considers these bodies as consisting of 

 swarms of distinct meteorites, which are illuminated partly 

 by the light of the sun, but which also give out a light of 

 their own arising from the numerous and violent collisions 

 which are always taking place, especially near the nucleus 

 where the swarm is densest. The most remarkable fact 

 about this light is that discovered by Huggins, namely, 

 that its spectrum is identical with that of a hydrocarbon. 

 We are sorry, however, not to find in this volume any ex- 

 position of that theory of Prof Tait's concerning the deve- 

 lopment and manifestation of the tails of comets to which 

 Sir W. Thomson referred in his presidential address to the 

 British Association in 1871 as " Tait's beautiful sea-bird 

 analogy." A " tactic arrangement " of brickbats extending 

 over millions of miles would perhaps account for what at 

 present appears from a dynamical point of view most 

 paradoxical in the behaviour of comets' tails, but the 

 dynamical explanation of this tactic arrangement itself 

 seems still to remain as a desideratum of the theory. 



J. Clerk Maxwell 



DR. BALL ON SCREWS 

 The Theory of Screws j a Study in the Dynamics of a 

 Rigid Body. By R. S. BaU, LL.D., F.R.S., Andrews' 

 Professor of Astronomy in the University of Dublin 

 and Royal Astronomer of Ireland. (Dublin : Hodges, 

 Foster, and Co., 1876.) 



DR. BALL has published several papers of late years 

 on the subject here treated. The present volume 

 contains the substance of these papers re-cast, wth addi- 

 tional matter, and with a greatly improved terminology. 



It has been shown by the combined labours of Poinsot, 

 Chasles, and Mobius, that there is a perfect mathe- 

 matical identity between the composition of forces and 

 couples on the one hand, and of rotations and transla- 

 tions on the other. Every small movement of a rigid 

 body consists of rotation round a definite line combined 

 with sliding along it, in other words, consists of a twist on 

 a definite screw, and every system of forces applied to a 

 rigid body is reducible to a force along a definite line, 

 together with a couple round it. The force is the 

 analogue of the rotation, the couple is the analogue of 

 the translation, and the combined action of the force and 

 couple is called by Dr. Ball a wrench on a screw. In 

 each case the screw consists of the definite line (fixed in 

 position but unlimited in length), associated with a 

 definite length, called the ^itch, namely, the quotient 

 of the translation by the rotation, or of the couple 

 by the force. The amplitude of a twist is the mag- 

 nitude of the rotation; the intensity of a wrench is 

 the magnitude of the force. A twist, or its analogue a 

 wrench, is defined by six numbers, one of which may be 

 the amplitude or intensity, and the other five will then be 

 common to aU twists and wrenches on the same screw. 

 A screw is therefore defined by five numbers. 



When a body twists while acted on by a wrench, the 

 work done by the latter is the continued product of the 

 amplitude of the twist, the intensity of the wrench, and a 

 third factor called the vi>tual co-efficient of the two 

 screws. The virtual co-efificient is a symmetrical func- 

 tion of the two screws, and when it vanishes the two 

 screws are called reciprocal. In other words two screws 



are said to be reciprocal if a wrench on one of them does 

 no work in virtue of a twist on the other. 



Taking six screws at random in space, we can express 

 any twist as the resultant of six twists, one on each of 

 these screws, and the amplitudes of the six components are 

 called the screw co-ordinates of the resultant. Wrenches 

 can be expressed in the same way, intensity being substi- 

 tuted for amplitude. 



If we take five screws and combine arbitrary twists 

 upon them, we obtain an infinite number of resultant 

 twists on an infinite number of resultant screws. These 

 resultant screws constitute a screw-complex of the 5th 

 order ; and in like manner we may have complexes of 

 lower orders down to the 2nd. 



Given a complex of the 5 th order, one definite screw 

 can be found which is reciprocal to it, that is to say, 

 which is reciprocal to every screw contained in the com- 

 plex. One practical application of this theorem is, that 

 if a rigid body has freedom of the 5 th order (or one 

 degree of constraint), one definite screw can be found 

 such that a wrench upon it can do no work on the body. 

 Given any complex of the «th order, there is one definite 

 complex of the 6 - «th order that is reciprocal to it, in 

 the sense that every screw of the one complex is reci- 

 procal to every screw of the other. 



In general problems on the dynamics of a rigid body, 

 it is usually advantageous to select the six screws of 

 reference, so that each of them shall be reciprocal to the 

 other five. Such screws are called co-reciprocals. 



For example, in discussing the action of wrenches on 

 a body which has freedom of the 4th order (two degrees 

 of constraint), four of the co-reciprocals should be selected 

 from the complex which defines the freedom, and the 

 other two will then of necessity define that other complex 

 of the 2nd order which is reciprocal to it and includes 

 every wrench that can be exerted by the constraints. 

 When an applied wrench is resolved into its components 

 on these six co-reciprocals, the first four components deter- 

 mine the movement of the body, the other two being 

 completely destroyed by the reaction of the constraints. 



It is not to be imagined that a wrench applied to a free 

 body tends in general to make it twist on the same screw 

 on which the wrench lies. If, however, we take six screws 

 coinciding two and two with the principal axes at the 

 centre of mass, and having pitches + a, + b, + c, where 

 a, b, c denote the radii of gyration round these axes, these 

 six screws will possess the property in question — an im- 

 pulsive wrench on any cne of them produces an instan- 

 taneous twist on the same. Dr. Ball calls these the six 

 P>rtncipal screws of inertia for a free body, and he further 

 shows that a constrained body has a smaller number of 

 such screws, the defect from six being equal to the number 

 of degrees of constraint. 



Again, he shows that if a body has freedom of the ti\h. 

 order, and has a position of stable equilibrium under the 

 action of forces which have a potential, n screws (called 

 harmonic screws) can be found possessing the following 

 property, viz., that the body can execute small oscillations 

 on any one of them, and will execute such oscillations if 

 it receive an arbitrary displacement and initial twist-velo- 

 city upon it. Also that any possible small oscillations of 

 the body can be resolved into n independent oscillations 

 on the harmonic screws. 



