133, 134] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 121 



of u e upon the several screws of the cylindroid ( 128). The representative 

 circle ( 50) will give a convenient geometrical construction. 



Let 6 l and 0. 2 be the two co-ordinates of 6 relatively to any two screws 

 of reference on the cylindroid. Then the components of the twist velocity 

 will be 00i and 00.,. The actual velocity of any point of the body will 

 necessarily be a linear function of these components. The square of the 

 velocity will contain terms in which 3 is multiplied into 0f, 0^0.,, #./, respec 

 tively. If, then, by integration we obtain the total kinetic energy, it must 

 assume the form 



whence, from the definition of u g 



The three constants, X, /m, v, are the same for all screws on the cylindroid. 

 They are determined by the material disposition of the body relatively to 

 the cylindroid. 



We have taken the two screws of reference arbitrarily, but this equation 

 can receive a remarkable simplification when the two screws of reference 

 have been chosen with special appropriateness. 



Fig. 18. 



Let the lengths AX and BX (fig. 18) be denoted by p l and p 2 , and if e 

 be the angle subtended by AB, we have from 57, 



X/?! 2 + 2/zp^ + vpJ - u e ~ (ps - Zptfz cos e + p./) = 0. 



Let us now transform this equation from the screws of reference A, B 

 to another pair of screws A , B . Let p^, p., be the distances of X from 

 A , B , respectively ; then, from Ptolemy s theorem, we have the following 

 equations : 



pl .A B = p.;.AA - pl .AB , 



p,.A B = p 2 .A B-p, .BB . 



