DYNAMICS OF A RIGID BODY. 125 



the pitches of three co-reciprocal screws is constant. This 

 theorem will be subsequently generalised ( 136). 



1 13. Screws of the Complex parallel to a Plane. Up to 

 the present we have been analysing the screw complex 

 by classifying the screws into groups of constant pitch. 

 Some interesting features will be presented by adopting 

 a new method of classification. We shall now divide 

 the general system into groups of screws which are 

 parallel to the same plane. 



We shall first prove that each of these groups con- 

 stitutes a cylindroid. For suppose a screw of infinite pitch 

 normal to the plane, 'then all the screws of the group 

 parallel to the plane are reciprocal to this screw of 

 infinite pitch. But they are also reciprocal to any three 

 screws of the original reciprocal system ; they, therefore, 

 form a screw complex of the second order ( 46) that is, 

 they constitute a cylindroid. 



We shall prove this in another manner. 



A quadric containing a line must touch every plane 

 passing through the line.* The number of screws of the 

 complex which can lie in a given plane is, therefore, 

 equal to the number of the quadrics of the complex 

 which can be drawn to touch that plane. 



The quadric surface whose equation is 



touches the plane Px + Qy + Rz + S = o, when the fol- 

 lowing condition is satisfied :f 



> - k] (A -*) + C(A - *) (A - 



* Salmon's Analytic Geometry of Three Dimensions, p. 74. 

 t Salmon, loc. cit., p. 49. 



