June 5, 1890] 



NATURE 



129 



there are oo^ screws in existence. Having established 

 the notion of these co-ordinates, there are next given, in 

 chapter v., expressions in terms of the co-ordinates for 

 the resultant of a number of wrenches or twists, for the 

 work done by a wrench on one screw during a twist on 

 another, and so on. These expressions are much simpli- 

 fied by selecting the screws of reference in a particular 

 manner, viz. so that any two of them are reciprocal, and 

 such a system of " co-reciprocal " screws is afterwards 

 always used. 



The expression for the virtual coefficient of two screws 

 is in general a lineo-linear function of the co-ordinates of 

 both screws. But this is simplified for the special system 

 of co-ordinate screws just mentioned, in reducing to an 

 expression of six terms only, each being the product of 

 the co-ordinates of the two screws relating to the same 

 co-ordinate screw into the parameter of this screw. This 

 expression must vanish if the two screws shall be reci- 

 procal. Hence the condition that a screw shall be 

 reciprocal to a given screw is expressed as a linear 

 equation between its co-ordinates, and it is important 

 to add that every linear equation between its co- 

 ordinates can be interpreted as meaning that the 

 screw is reciprocal to some other screw. But one 

 linear equation enables us to express one of the 

 co-ordinates in terms of the others, so that all the 

 co-ordinates of all screws which are reciprocal to a given 

 screw can be expressed in terms of five co-ordinates, in 

 other words, a screw in a complex of order five is determined 

 by five co-ordinates. In the same way two linear equations 

 limit a screw to a complex of order four, and so on, till 

 we come to five equations as determining one single 

 screv/ ; which also shows that there is always one screw 

 which is reciprocal to five given screws. 



We leave for the moment the line followed by Ball and 

 Gravelius, in order to indulge in some very general specu- 

 lations, in close connection with chapter xix., which seem 

 best suited to give, in as short a compass as possible, a 

 clear insight into the nature of the whole system of 

 screws. 



We are accustomed to express the fact that the number 

 of points in a plane is cxs'^ by saying a plane, or in fact 

 any surface, is of two dimensions if we consider the points 

 as elements. Space is, in the same sense, of three dimen- 

 sions, whilst it is of four dimensions if we consider the 

 lines as elements. 



We may extend this language, and say the aggregate 

 of all screws forms a space of five dimensions, or as 

 Clifford would have said, it is a five-way spread. If we 

 now assume between the co-ordinates one equation, we 

 may speak of the locus of screws whose co-ordinates 

 satisfy this equation. It will be a four-way spread, or a 

 space of four dimensions. This locus is called by Ball a 

 screw-complex of order five and degree w, if in is the 

 degree of the equation. The complexes spoken of before 

 are of the first degree. 



The geometrical theory of screws becomes thus identical 

 with the geometry of a space of five dimensions, which 

 latter we may call the screw- space. 



Let us consider now two such complexes of ist degree, 

 one of order m, the other of order n. The first is de- 

 termined by a set of 6 - m linear equations between the 

 co-ordinates, the second by 6 - ?i such equations. All 

 screws common to both have therefore to satisfy 

 12 — 7/1 — n equations, and in case that this number is 

 not greater than six, they will constitute a complex of 

 order 6 - (12 - ?« — «) = m -{■ n - 6. Thus a complex 

 of order 4 and a complex of order 5 will have a complex 

 of order 3 in common, whilst two complexes of order 3 

 will in general have no screw in common, though 

 they may have a single screw or a whole cylindroid in 

 common. 



The geometrical theory of screws as the geometry of a 

 particular space of five dimensions is not a mere ex- 



NO. 1075, '^'OL. 42] 



tension of the ordinary Euclidian geometry. The six 

 homogeneous co-ordinates of a screw are, as has already 

 been mentioned, connected by an equation. This is of 

 the form R = i, where R is a quadratic expression of the 

 co-ordinates. All elements at infinity in our screw-space 

 are given by the equation 1=0 or by R = o. The 

 absolute is thus a quadric locus, and therefore we have to 

 deal with non-Euclidian geometry. 



The advantage to the theory of screws to be derived 

 from a study of this geometry are apparent at every step. 

 We may in our screw-space conceive curves and 

 surfaces of from i to 4 dimensions, by taking one or 

 more equations between the co-ordinates. Of these, 

 equations of the first degree determine the screw- 

 complexes. But equations of the second degree, which 

 determine quadric complexes, or as Ball calls them screw- 

 complexes of second degree, are also constantly of use. 

 Such an equation may be taken in a complex of order n. 

 In the treatise before us they appear in congruences of the 

 3rd and 6th order. We will give here one illustration. 



Let/i, p<,, . . . pn be the pitches of the n co-reciprocal 

 co-ordinate screws, and let a^, a.^, . . . an be the co- 

 ordinates of a screw a with pitch ^a- Then is/a given 

 by the equation 



: /lOi^ + /.^a^" + 



+ Pn<'n'. 



This equation can be made homogeneous by aid of 

 R = I, and becomes 



RA =/!«!- -I- A«2'' + . . . . + Aa,A 



where R also is supposed to contain n of the a only, the 

 others being replaced by aid of the linear equations which 

 determine the complex of order n. It follows that the 

 absolute R = o is the locus of screws of infinite pitch, 

 whilst 



/l«l" + A«2' +....+ pnO.,? = O 



is the locus of screws of zero-pitch. Both are quadrics. 



If we take a screw ^, we may form its polar with regard 

 to any quadric. If we select the last quadric mentioned, 

 the polar is 



/lOj/Si -H AooiSa + • 



-i- pnOLn^n 



But this equation is also the condition that a and /3 are 

 reciprocal screws. In each complex the quadric of zero- 

 pitch becomes thus of special importance, reciprocal 

 screws being conjugate poles with regard to them. 



As we cannot directly realize a space of more than 

 three dimensions, it becomes of importance to represent 

 the elements in such a space by other elements in 

 ordinary space, and, when possible, by elements in a 

 plane. That this is always possible is clear. 



For instance, as all conies in a plane are oo' in number, 

 we have as many conies in a plane as there are screws in 

 space, and we may therefore represent each screw by a 

 conic in a plane. To screws on a cylindroid would then 

 correspond all conies in a pencil. We might then speak 

 of the cross-ratio of four screws as given by the cross- 

 ratio of the corresponding conies in the pencil. All 

 screws belonging to a complex of order 3 would be 

 represented by conies forming a net, i.e. by conies having 

 a common polar triangle. 



We thus get a graphical representation in a plane, and 

 can obtain our results by constructions in a plane. But 

 the geometry of conies in a plane has scarcely been far 

 enough developed to make general use of them, and for 

 screw-complexes of lower order simpler representations 

 may be found. Thus the screws on a cylindroid can be 

 represented most conveniently by the points on a circle 

 which stands in close relation to the cylindroid and gives 

 rise to a graphical solution of problems relating to a body 

 with two degrees of freedom. This is done in chapter xx., 

 full of interesting detail. Again, screws in a complex of 

 order 3, whose number is 00 ^^ can be represented by 



