401] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 441 



first time. No doubt we find the intervene to be the logarithm of an an- 

 harmonic ratio of four quantities, but these quantities are not distances 

 nor are they quantities homologous with the intervene. They are simply 

 numerical. The four numbers, X, jj,, X , X&quot; are merely introduced to define 

 four objects, one of them being, 



and the others are obtained by replacing X by //,, X , X&quot;, respectively. All 

 we assert is, that if we choose to call the two objects defined by X and X&quot; 

 the objects at infinity, and that if we desire the intervene between the 

 objects X and /j, to possess the properties that we have already specified, 

 then the only function possible will be the logarithm of the anharmonic ratio 

 of these four numbers. 



The word anharmonic is ordinarily applied in describing a certain 

 function of four collinear points. In the more general sense, in which we 

 are at this moment using the word, it does not relate to any geometrical or 

 spacial relation whatever ; it is a purely arithmetical function of four abstract 

 numbers. 



We may also observe that the relation between 6 and the intervene 8 

 is given by the equation 



8 



P H , 1 

 /i i/v \ \ 



&quot;It* ~ x )-s- 



and the expression of the intervene as a function of 6 ; that is, the expression 

 F(6} is 



401. Another process. 



We may also proceed in the following manner. Let us denote the values 

 of X for the infinite objects on the range by pe ie and pe~ ie . 



If then X, fji be two parameters for two objects at an intervene , we must 

 have (p. 439) 



X/i + X (e p cos 6) + fM ( e - p cos 6) + p 2 = 0. 



Solving for e, we have 



_ X/i p cos 6 (X + p,} + p z 

 fji X 



The intervene 8 must be some function of e, whence 



