391] THE THEORY OF PERMANENT SCREWS. 429 



If the restraints are such that the number of degrees of freedom is less 

 than four, then an infinite number of rigid bodies can be designed, such 

 that the impulsive screws and their corresponding instantaneous screws 

 shall be represented by a given chiastic homography. If n exceed four then 

 it will not in general be possible to design a rigid body such that its corre 

 sponding impulsive screws and instantaneous screws shall agree with a given 

 chiastic homography. If, however, n 4 then it is always possible to design 

 one but only one rigid body so that its pairs of corresponding impulsive 

 screws and instantaneous screws shall be represented by a given chiastic 

 homography. 



Returning to the three-system we may remark that, having settled the 

 inertia conic in the plane representation we are not at liberty to choose 

 three arbitrary points as representing the three permanent screws. For 

 if these three points were to be chosen quite arbitrarily, then six relations 

 among the co-ordinates of the rigid body would be given, and the conic of 

 inertia would require five more conditions. Hence the co-ordinates of the 

 rigid body would have in general to satisfy eleven conditions which, of 

 course, is not generally possible, as there are only nine such co-ordinates. 

 It is therefore plain that when the conic of inertia has been chosen at least 

 two other conditions must necessarily be fulfilled by the three points which 

 are to represent the permanent screw. This fact is not brought out by 

 the method of 389 in which, having chosen the three permanent screws 

 arbitrarily, we have then written down the general equation of a conic 

 as the inertia conic. This conic should certainly fulfil at least two con 

 ditions which the equations as there given do not indicate. 



We therefore calculate directly the expression for the kinetic energy of 

 a body in the position #/, # 2 , 3 twisting about a screw with twist velocities 

 #1, 6 2 , 6 3 when the screws of reference are the three principal screws of the 

 three-system with pitches a, b, c, and when a? , y , z , If, 1 2 2 , 1 3 2 , pf, p 2 * } p/ are 

 the nine co-ordinates ( 324) of the rigid body relative to these axes. 



It is easily shown that we have for the kinetic energy the mass M of the 

 body multiplied into the following expression where squares and higher 

 powers of #/, # 2 &amp;gt; QS are omitted : 



i (a 2 ^ 2 + W + c 2 3 2 + p*0* + p*ej + p 3 *0 3 *) 



+ 0A (c -b)x + 0A (a -c)y + 0A (b - a) z. 



+ 0A(p* 2 - p&amp;gt;? + uc - ab) 

 + 0A (V - cz ) - 6 A (4 2 



