450 THE THEORY OF SCREWS. 



An object L will be defined by an equation of the form, where L ly L 2 , L 3 



are numerical, 



L 3 x 3 = 0, 



for any two sets of values, x lt x z , x 3 , which satisfy this equation, will deter 

 mine a pair of ranges which have the required object in common. 



Let the relation between the co-ordinates of an infinite range be 



/Oi, a? 2 , s ) = 0; 



then the infinite ranges in the star, whose vertex is the object L, will be 

 defined by co-ordinates obtained from the simultaneous solution of 



= 0, 



But there can only be two such ranges ; and, accordingly, the latter of these 

 equations must be of the second degree. We hence deduce the following 

 important result : 



The infinite ranges in an extent may be represented by the different groups 

 of values of the co-ordinates x ly x z , x 3 which satisfy one homogeneous equation 

 of the second degree. 



Remembering that the existence of zero intervene between every pair of 

 objects on a range is a consequence of the coincidence of the two objects of 

 infinite intervene on that range, we have the following result : 



The range through, two consecutive objects of infinite intervene is a range of 

 zero intervene. 



And, similarly, we have the following : 



The object common to two consecutive ranges of infinite departure is the 

 vertex of a star of zero departure. 



411. Relations between Departure and Intervene. 



With reference to the theory of Departure, we thus see that there is a 

 system of critical ranges, and a system of critical objects in each extent. 

 Every critical range has an infinite departure from every other range. Every 

 critical object is the vertex of a star of which the departure between every 

 pair of ranges is zero. It will be remembered that in the theory of the 

 intervene we were conducted to the knowledge that every extent contained 

 a system of objects and ranges, critical with regard to the intervene. Each 

 critical object had an infinite intervene with every other object in the 

 extent. Each critical range possessed the property that the intervene 

 between any two objects thereon is zero. There are, thus, objects and 

 ranges critical with repect to the intervene. There are also objects and 

 ranges critical with respect to the departure. But nothing that we have 



