128 



NATURE 



[June 5, 1890 



droid which meet at right angles. Let the corresponding 

 screws be o and /3. Then if the circle when in a plane 

 with a be turned about its diameter through a right angle 

 it will be parallel to the plane of the pencil and may be 

 taken to coincide with it. In this position we get the circle 

 used in chapter xx. We recommend the reader to go 

 through the first pages of this chapter when reading the 

 third and fourth. 



To combine two wrenches on two screws, a and /3, 

 we have to construct the cylindroid containing the screws 

 and the flat pencil spoken of. If on the two rays in this 

 pencil which are the projections of a and ^ the intensities 

 of the wrenches be set ofif (they are the forces which to- 

 gether with couples constitute the wrenches), then their 

 resultant gives not only the intensity of the resultant 

 wrench, but it lies on the ray which is the projection of 

 the screw of the resultant wrench. From this follows at 

 once : Any two wrenches on screws of a cylindroid com- 

 bine to a wrench whose screw lies again on the cylindroid ; 

 and conversely, a wrench on a screw belonging to a cylin- 

 droid can be decomposed into two wrenches on any two 

 given screws on the cylindroid. Also, on any three screws 

 of a cylindroid wrenches can be determined which are in 

 equilibrium. It need scarcely be stated that the ratios 

 only of their intensities are determined ; but it is of im- 

 portance to remember this. 



The above results for the composition of wrenches hold 

 also for twists about screws, provided that their amplitudes 

 are very small, in conformity with the well-known fact 

 that small rotations are combined in the same manner 

 as forces. For this reason Sir R. Ball has limited his 

 investigations to cases where the twist velocities have 

 infinitely small amplitudes. These include equilibrium, 

 beginnings of motion due to impulses and small oscilla- 

 tions. He also supposes the forces always to have a 

 potential. Within these limits his results are of absolute 

 generality. 



The remarkable analogy between forces and rotations 

 which appears in analytical mechanics rather as an 

 accidental, though interesting, circumstance, is raised in 

 the theory of screws to a principle of paramount im- 

 portance. 



If a rigid body acted on by a wrench receives a small 

 twist, then the work done by the wrench is the product of 

 the intensity of the wrench, of the amplitude of the twist, 

 and of a geometrical factor which depends solely upon 

 the two screws of the wrench and twist. Half this factor 

 Ball calls " tlie virtual coeffi.de7tt of the two screws." If 

 the screws meet it is proportional to the cosine of the 

 angle between them ; if the pitches of both screws vanish, 

 or more generally if their sum vanishes, it becomes the 

 moment of the two lines on which the screws lie. It par- 

 takes, therefore, of the nature of both these quantities, 

 and its analogies to the cosine especially are, in many 

 cases, very marked. If the virtual coefficient vanishes, 

 then no work is done by the wrench in consequence of 

 the twist. Now the virtual coefficient of two screws, 

 a and jS, depends symmetrically on both, hence if a wrench 

 on a does no work when the body is displaced by a twist 

 about /3, then also a wrench on /3 does no work during a 

 twist on o. For this i-eason two screws whose virtual 

 coefficient vanishes are called reciprocal. 



An immediate consequence of the definition of reciprocal 

 screws is this, that a screw which is reciprocal to two 

 screws, a, /3, is reciprocal to all screws on the cyhndroid 

 determined by a, ^. For a twist about any screw, 7, on the 

 cylindroid can be decomposed into two about a and ^ ; but 

 the wrench can do no work against these, and therefore it 

 can do no work against a twist about y. 



It is also proved that through every point in space 

 there pass a single infinite number of screws, which are 

 reciprocal to a cylindroid. These lie on the generators 

 of a cone of the second order. Similarly, all screws in a 

 plane which possess the property in question envelop a 



NO. 1075, VOL. 42] 



conic, and in chapter xxi. it is shown that this is always 

 a parabola. 



Two screws which meet can be reciprocal only if 

 they meet at right angles or if the sum of their pitches 

 vanishes. This gives rise to one of the most powerful 

 methods for finding reciprocal screws. Thus, as every 

 line meets a cylindroid in three points, and there- 

 fore cuts three screws on it, and as the cylindroid 

 contains only two screws of equal pitch, it follows 

 a screw, a, reciprocal to a cylindroid must cut one 

 screw on it at right angles, and the two others which it 

 meets must have equal pitches, viz. these must be equal 

 and opposite to the pitch of a ; and from this, again, it is 

 easily deduced that every line which meets one screw on 

 a cylindroid at right angles cuts, besides, two others 

 which have equal pitch ; for if on this line a screw be 

 taken with a pitch equal to one of the two remaining 

 screws which it cuts, it will be reciprocal to the 

 cylindroid. 



Just as two wrenches on screws a and iS always combine 

 to a wrench on a screw lying on a certain cylindroid, so 

 three wrenches on screws a, /3, y, which do not lie on a 

 cylindroid, combine to a wrench on a fourth screw which 

 is connected with the three given ones, and which de- 

 pends on the two ratios only of the intensities of the 

 three given wrenches. 



The entirety of all the screws which are got by varying 

 these ratios forms a system of a double infinite number of 

 screws, which has been called a screw-complex of the 

 third order. 



If any four screws belonging to such a complex are 

 selected, then a wrench on one of them can be decom- 

 posed into three wrenches on the others. It is also 

 always possible to determine wrenches on the four screws 

 which are in equilibrium, and the ratios of their intensi- 

 ties alone are then determined. Similarly, five indepen- 

 dent screws, i.e. screws which do not belong to a complex 

 of lower order, give rise to a complex of order five, and 

 six independent screws to a complex of order six. To 

 this latter complex all screws in space belong, for in 

 chapter v. it is shown that in general any wrench can be 

 decomposed into six wrenches on six arbitrarily selected 

 screws. A screw-complex of order two is nothing but 

 a cylindroid, and a complex of order one consists of 

 one single screw. That a complex of order six exhausts 

 all screws in space, says only that the number of all 

 screws is 02'', if oo^ denotes the number of points in a line^ 

 or the number of values which a single real variable, x, 

 may assume. That the number of all screws is co-' is also 

 at once evident if we consider that the number of lines 

 in space is oo*, and that on each line we have a single 

 infinite number of screws which are obtained by giving 

 its pitch all possible values from - 00 to + 00. 



There is an important theorem that the screws which 

 are reciprocal to all screws in a complex of order n form 

 themselves a complex of order d-n. 



One of the chief uses made of these results consists in 

 the introduction of screw co-ordinates, viz. six indepen- 

 dent screws are selected as co-ordinate screws. Then 

 the intensities of the components of a wrench on these 

 six screws are taken as the co-ordinates of the wrench. 

 In the same way the co-ordinates of a twist are obtained. 

 Lastly, by the co-ordinates of a screw are understood 

 the co-ordinates of a wrench of unit intensity on the 

 screw, or those of a twist of unit amplitude about it. 

 To get, then, the co-ordinates of any wrench on, or a 

 twist about, a screw, the co-ordinates of the latter have 

 only to be multiplied by the intensity of the wrench or 

 the amplitude of the twist. Between these screw co-ordin- 

 ates exists, however, an equation of the second degree, 

 just as between the ordinary homogeneous point co- 

 ordinates there exists a linear equation. A screw is 

 thus completely detennined by the ratios of its six co- 

 ordinates, i.e. by five numbers, which again shows that 



