182 Prof. R. S. Ball's Description of the "Cylindroid." 



If, after all the screws have been tried, the body be found ca- 

 pable of displacement about one screw only, the body possesses 

 the lowest degree of freedom. If one screw (A) be discovered, 

 and, the trials being continued, a second screw (B) be found, the 

 remaining trials may be abridged by considering the information * 

 which the discovery of two screws affords. 



The body may receive any displacement about one or both 

 of the two screws A and B. The composition of these displace- 

 ments gives a resultant which could have been produced by dis- 

 placement about a single screw. The locus of this single screw 

 is the conoidal cubic surface which has been called the " cylin- 

 droid " (at the suggestion of Professor Cayley) . The equation of 

 the surface is 



z(%2 + if)-2accy = Q. 



Any line (s) upon the surface is considered to be a screw, 

 of which the pitch is 



c + a cos 26, 



where c is any constant, and 6 is the angle between s and the line 



y=o, 



z=0. 



The fundamental property of the cylindroid is thus stated. If 

 any three screws of the surface be taken, and if a body be dis- 

 placed by being screwed along each of these screws through a 

 small angle proportional to the sine of the angle between the 

 remaining screws, the body after the last displacement will oc- 

 cupy the same position that it did before the first. 



The equation of the cylindroid is thus deduced. Take as 

 axes of x and y two screws intersecting at right angles whose 

 pitches are c-\-a and c— a ; a body is rotated about each of these 

 screws through angles co cos 6, co sin 6 respectively. The corre- 

 sponding translations are (c + a) co cos 6 , and (c — • a) co sin 6. 

 The resultant of the translations may be resolved into two com- 

 ponents, of which (c + a cos 26) co is parallel to the resultant of 

 the rotations, and aco sin 26 is perpendicular to the same line. 

 The latter component has merely the effect of transferring in a 

 normal plane the resultant of the rotations to a distance a sin 26, 

 the resultant moving parallel to itself. The two original screw- 

 movements are therefore compounded into a single screw whose 

 pitch is c -f a cos 26. The position of the screw is defined by the 

 equations 



^y — x tan 6, 



_z = asin.26. 



{: 



Eliminating 6, we have the equation of the surface. 



