THE THEORY OF SCREWS. [182- 



182. Four Screws of the Screw System. 



Take any four screws a, ft, 7, 8 of the screw system of the third order. 

 Then we shall prove that the cylindroid (a, ft) must have a screw in common 

 with the cylindroid (7, &). For twists of appropriate amplitudes about a, 

 ft, 7, B must neutralise, and hence the twists about a, ft must be counter 

 acted by those about 7, & ; but this cannot be the case unless there is 

 some screw common to the cylindroids (a, ft) arid (7, 8). 



This theorem provides a convenient test as to whether four screws 

 belong to a screw system of the third order. 



183. Geometrical notes. 



The following theorem may be noted : 



Any ray 77 which crosses at right angles two screws a, ft of a three-system 

 is the seat of a screw reciprocal to the system. 



For, draw the cylindroid a, ft, then of course 77, whatever be its pitch, 

 is reciprocal to all the screws on this cylindroid. Through any point P on 

 77 there are two screws of the system which lie on the cylindroid, and there 

 must be a third screw 7 of the system through P, which, certainly, does 

 not lie on the cylindroid. If, therefore, we give 77 a pitch p y) it must be 

 reciprocal to the three-system. 



In general, one screw of a three-system can be found which intersects 

 at right angles any screw ivhatever 77. 



For 77 must, of course, cut each of the quadrics containing the screws 

 of equal pitch in two points. Take, for example, the quadric with screws 

 of pitch p. There are, therefore, two screws, a and ft of pitch p belonging 

 to the system, which intersect 77. The cylindroid a, ft must belong to the 

 system, and from the known property of the cylindroid the ray ij, which 

 crosses the two equal pitch screws ( 22), must cross at right angles some 

 third screw 7 on this cylindroid ; but this belongs to the three-system, and 

 therefore the theorem has been proved. 



184. Cartesian Equation of the Three-System. 



If we are given the co-ordinates of any three screws of a three-system 

 with reference to six canonical co -reciprocals, we can calculate in the 

 following manner the equation to the family of pitch quadrics of which the 

 three-system is constituted. 



Let the three given screws be a, ft, 7, with co-ordinates respectively 



