159] THE GEOMETRY OF THE CYLINDROID. 147 



screw e not on S ; for as e and any screw 7 on S are reciprocal to lt # a , 3 ,0 4 , 

 it will follow that any screw on the surface made from e and 7, just as S is 

 made from a and ft, must also be reciprocal to 6 l) 2 , 3 , 4 . As 7 may be 

 selected arbitrarily on S, we should thus find that the screws reciprocal to 

 &i, &2, $s&amp;gt; #4 were not limited to one surface, but constituted a whole group of 

 surfaces, which is contrary to what has been already shown. It is therefore 

 the same thing to say that a screw lies on S, as to say that it is reciprocal to 

 0i, 6* d. ( 24). 



Since the condition of reciprocity involves the pitches of the two screws 

 in an expression containing only their sum, it follows that if all the pitches 

 on 1} 2 , 3 , # 4 be diminished by any constant in, and all those on S be 

 increased by m, the reciprocity will be undisturbed. Hence, if the pitches 

 of all the screws on S be increased by + m, the surface so modified will still 

 retain the property, that twists about any three screws will neutralize each 

 other if the amplitudes be properly chosen. 



We can now show that there cannot be more than two screws of equal 

 pitch on 8 ; for suppose there were three screws of pitch m, apply the 

 constant m to all, thus producing on S three screws of zero pitch. It must 

 therefore follow that three forces on 8 can be made to neutralize ; but this is 

 obviously impossible, unless these forces intersect in a point and lie on a 

 plane. In this case the whole surface degrades to a plane, and the case is a 

 special one devoid of interest for our present purpose. It will, however, be 

 seen that in general S does possess two screws of any given pitch. We can 

 easily show that a wrench can always be decomposed into two forces in such 

 a way that the line of action of one of these forces is arbitrary. Suppose 

 that 8 only possessed one screw A, of pitch m. Reduce this pitch to zero ; 

 then any other wrench must be capable of decomposition into a force on X 

 (i.e. a wrench of pitch zero), and a force on some other line which must lie 

 on S; therefore in its transformed character there must be a second screw 

 of zero pitch on S, and, therefore, in its original form there must have been 

 two screws of the given pitch m. 



Intersecting screws are reciprocal if they are rectangular, or if their 

 pitches be equal and opposite ; hence it follows that a screw 6 reciprocal to 

 S must intersect 8 in certain points, the screws through which are either at 

 right angles to or have an equal and opposite pitch thereto. 



From this we can readily show that S must be of a higher degree than 

 the second ; for suppose it were a hyperboloid and that the screws lay on 

 the generators of one species A, a screw which intersected two screws 

 of equal pitch m must, when it receives the pitch m, be reciprocal to the 

 entire system A. We can take for one of the generators on the hyper 

 boloid belonging to the species B ; will then intersect every screw of the 



10-2 



