TRANSACTIONS OF THE SECTIONS. 27 



to s lie upon a cylindroid, the principal plane of ■which is called the reciprocal 

 jdanc. Then the required instantaneous screw u is detei-mined ; for it is parallel to 

 that diameter of the ellipsoid of equal kinetic energy which is conjugate to the 

 reciprocal plane. 



Ihe demonstration is as follows: — Any three conjugate diameters of the ellip- 

 soid of equal kinetic energy are pai-allel to three screws of the system, which are 

 conjugate scretcs of kinetic energy. The property possessed by three conjugate 

 screws of kinetic energy A, B, C, is that if A', B', C be three impulsive screws 

 corresponding respectively to A, B, C as instantaneous screws, then A' is reciprocal 

 to B and C, B' is reciprocal to A and C, C is reciprocal to A and B. 



If u be one of three conj ugate screws of kinetic energy, the two others must be 

 parallel to the reciprocal plane, and therefore reciprocal to s. Hence an impulsive 

 wrench about s must make the body commence to twist about u, 



II. 



The same construction may be arrived at in a different manner. 

 Let q be the screw of the coordinate-system which is normal to the plane reci- 

 procal to s. 



MV 

 LetF,= — — ' be the impulsive wrench which acts about s for the infinitel}' 



small time t. 



Let £0^ be the twist velocity with which a body must twist uniformly round q 

 in order to do one unit of work against F in the time t. 



Draw a plane parallel to the reciprocal plane at a distance a>q from the kinematic 

 centre. 



Draw the cone from the kinematic centre to the intersection of this plane with 

 the ellipsoid of equal kinetic energy. 



Then all the screws of the coordinate-system which are parallel to the gene- 

 rators of this cone possess the following property : — That if the body be constrained 

 to twist about any one of these screws it will, in consequence of the impulsive 

 wrench F^, move oft' from rest with the unit of kinetic energy. 



The screw s being given, F^ will vary inversely as a> ; consequently when the 

 plane touches the ellipsoid, and when the cone has shrunk to one right line, a 

 smaller impulse about s will give the body the unit of kinetic energy about the 

 screw of the system parallel to that line, than if the body had been constrained 

 about any other screw of the system. 



Applying Euler's theorem, that a body will always move off" with the maximum 

 kinetic energy, -we arrive at the construction already given. 



III. 



Converseh', given the instantaneous screw u, about which the body will com- 

 mence to tvsdst, selected from the general coordinate-system with three degi-ees of 

 freedom, determine the corresponding impulsive screw s. 



This problem is really indeterminate ; the conditions to be fulfilled by s are 

 thus proved. Draw the plane in the ellipsoid of equal kinetic energy, conjugate 

 to the direction of «<. Construct the cylindroid of screws belonging to the system 

 which are parallel to this plane, then s may be any screw reciprocal to this cylin- 

 droid. For example, through any point a cone of screws can be drawn, any one 

 of which, as an impulsive screw, corresponds to u as an instantaneous screw. 



Contributions to the Theory of Screws. 

 By RoBEET Stawell Ball, LL.D., F.R.S. 



1. Coordinates uf a Screw. — Six screws, each of which is reciprocal to the re- 

 maining five, are called a group of coreciprocals *. If the unit twist velocity about 



* A group of six coreciprocals is intimately connected with Ihe group of six funda- 

 mental com)ilesP8 already introduced into geometrv by Dr. Felix Klein (see ' Math. Ann ' 

 Band ii. p. 203). 



