256 THE THEORY OF SCREWS. [236 



Let the radius vector of length p g + a = R be marked off along each screw 

 6 drawn through P, then the equation becomes 



or squaring (Z 2 + F 2 + Z*) X* = h*Z\ 



This represents a surface of the fourth degree. A model of the surface 

 has been constructed*. It is represented in Fig. 41, and from its resemblance 

 to the valves of a scallop shell the name pectenoid is suggested. 



The geometrical nature of a pectenoid is thus expressed. Given a screw 

 a of pitch p a and a point situated anywhere. If a screw 6 drawn through 

 be reciprocal to a then the extremity of a radius vector from along 6 equal 

 to p a + p e will trace out a pectenoid. 



All pectenoids are similar surfaces, they merely differ in size in accordance 

 with the variations of the quantity h. The perpendicular from upon a is 

 a nodal line, and this is the only straight line on the surface. The pectenoid 

 though unclosed is entirely contained between the pair of parallel planes 

 Z = + h, Z = h. Sections parallel to the plane of Z are hyperbolas. Any 

 plane through the nodal line cuts the pectenoid in a circle. 



A straight wire at right angles to the nodal line marked on the model 

 indicates the screw reciprocal to the five-system. A second wire starts 

 from the origin and projects from the surface. It is introduced to show 

 concisely what the pectenoid expresses. If this wire be the axis of a 

 screw 6 whose pitch when added to the pitch of the screw a, is equal to 

 the intercept from the origin to the surface then the two screws are 

 reciprocal. The interpretation of the nodal line is found in the obvious 

 truth that when two screws intersect at right angles they are reciprocal 

 whatever be the sum of their pitches. One of the circular sections made 

 by a plane drawn through the nodal line is also indicated in the model. The 

 physical interpretation is found in the theorem already mentioned, that all 

 screws of the same pitch drawn through the same point and reciprocal to 

 a given screw will lie in a plane. 



With the help of the pectenoid we can give another proof of the theorem 

 that all the screws of a four-system which can be drawn through a point lie 

 on a cone of the second degree ( 123). 



Let be the point and let a and /3 be two screws on the cylindroid 

 reciprocal to the system. Let 6 be a screw through belonging to the 

 four-system and therefore reciprocal to a and to /9. 



Then for the pectenoid relating to and a, we have 



(p e + pa )M-kN=0, 

 where M=0, N=0 represent planes passing through 0. 



* Transaction* of the Royal Irish Academy, Vol. xxv. Plate xn. (1871). 



