20 THE THEORY OF SCREWS. [13, 



The conoidal cubic surface represented by this equation has been called 

 the cylindroid*. 



Each generating line of the surface is conceived to be the residence of a 

 screw, the pitch of which is determined by the expression 



When a cylindroid is said to contain a screw, it is not only meant that the 

 screw is one of the generators of the surface, but that the pitch of the screw 

 is identical with the pitch appropriate to the generator with which the screw 

 coincides. 



We shall first show that it is impossible for more than one cylindroid to 

 contain a given pair of screws 6 and &amp;lt;f&amp;gt;. For suppose that two cylindroids 

 A and B could be so drawn. Then twists about 6 and &amp;lt;f&amp;gt; will compound 

 into a twist on the cylindroid A and also on the cylindroid B ( 14). There 

 fore the several screws on A would have to be identical with the screws on B, 

 i.e. the two surfaces could not be different. That one cylindroid can always 

 be drawn through a given pair of screws is proved as follows. 



Let the two given screws be 6 and &amp;lt;, the length of their common perpen 

 dicular be h, and the angle between the two screws be A ; we shall show that 

 by a proper choice of the origin, the axes, and the constants p a and pp, a 

 cylindroid can be found which contains 6 and &amp;lt;. 



If I, m be the angles which two screws on a cylindroid make with the 

 axis of oc, and if z ly z*, be the values of z, we have the equations of 

 which the last four are deduced from the first six 



* This surface has been described by Pliicker (Neue Geometric des Eaumes, 1868-9, p. 97) ; he 

 arrives at it as follows : Let ft = 0, and ft = be two linear complexes of the first degree, then all 

 the complexes formed by giving /* different values in the expression O + /ufi = form a system of 

 which the axes lie on the surface z (x^ + y 2 ) - (k - k )xy = Q. The parameter of any complex of 

 which the axis makes an angle w with the axis of x is & = fc cos 2 w + fc sin 2 w. Pliicker also con 

 structed a model of this surface. 



Pliicker does not appear to have noticed the mechanical and kinematical properties of the 

 cylindroid which make this surface of so much importance in Dynamics ; but it is worthy of 

 remark that the distribution of pitch which is presented by physical considerations is exactly 

 the same as the distribution of parameter upon the generators of the surface, which Pliicker 

 fully discussed. 



The first application of the cylindroid to Dynamics was made by Battaglini, who showed that 

 this surface was the locus of the wrench resulting from the composition of forces of varying ratio 

 on two given straight lines (Sulla serie dei sistemi di forze, Eendic. Ace. di Napoli, 1869, p. 133). 

 See also the Bibliography at the end of this volume. 



The name cylindroid was suggested by Professor Cayley in 1871 in reply to a request which 

 I made when, in ignorance of the previous work of both Pliicker and Battaglini, I began to 

 study this surface. The word originated in the following construction, which was then 

 communicated by Professor Cayley. Cut the cylinder x i + y&quot;=(p -p^ in an ellipse by the 

 plane z x, and consider the line x = Q, y=P B ~P a - If any plane z~c cuts the ellipse in the 

 points A, B and the line in C, then CA, CB are two generating lines of the surface. 



