Prof. R. S. Ball's Description of the « Cylindroid." 183 



The property of the surface is thus proved. Let 6 V 2 , 3 be 

 the angles of three screws upon the surface, and a> 1; &> 2 , co s be 

 the displacements about them. Each of these displacements 

 may be resolved into screw-displacements about the screws of x 

 and y.' The conditions necessary and sufficient for the displace- 

 ments to neutralize each other are 



o) 1 sin 6 1 -f <w 2 sin d + co 3 sm ^3— 0> 

 co 1 cos l + co 2 cos # 2 + 0) 3 cos #3=0. 



Thus each rotation is proportional to the sine of the angle 

 between the other two screws. 



For the complete determination of the cylindroid and the 

 pitch of all its screws, we must have the quantities a and c. 

 These quantities, as well as the position of the cylindroid in 

 space, are completely determined when two screws of the system 

 are known. 



In the model of the cylindroid which is exhibited, the para- 

 meter a is 2*6 inches. The wires which correspond in the model 

 with the generating lines of the surface represent the axes of the 

 screws. The distribution of pitch upon the generating lines is 

 shown by colouring a length of 26 x cos 20 inches upon each 

 wire. The distinction between positive and negative pitches is 

 indicated by colouring the former red and the latter black. 

 This model is in the possession of the London Mathematical 

 Society, 22 Albemarle Street. 



It is remarkable that the addition of any constant (c) to all 

 the pitches attributed in the model to the screws does not affect 

 the fundamental property of the cylindroid. 



When a rigid body is found capable of being displaced about 

 a pair of screws, it is necessarily capable of being displaced about 

 every screw on the cylindroid determined by that pair. 



The theorem of the cylindroid includes as particular cases the 

 well-known rules for the composition of two displacements pa- 

 rallel to given lines, or of two small rotations about intersecting 

 axes. If the parameter a be zero, the cylindroid reduces to a 

 plane, and the pitches of all the screws become equal. If the 

 arbitrary constant (c) which expresses the pitch be infinite, we 

 have the theorem for displacements ; and if the pitch be zero we 

 have the theorem for rotations. As far as the composition of 

 two displacements is concerned, the plane can only be regarded 

 as a degraded form of the cylindroid, from which the most essen- 

 tial feature has disappeared. 



Royal College of Science, 

 July 1871. 



