PHILLIPS AND MOORE. — LINEAR DISTANCE AND ANGLE. 57 



p cr represents a transformation which changes a space R compHmentary 

 to p into a space {R p) cr which is given by the locus of S in 



{R p) (S <J) = 0. 



Linear Distance and Angle in Three Dimensions. 



9. Linear distance between two points. We de fine the dis- 

 tance between two points A, B as such a function A B of their 

 coordinates that (1) if one is fixed the other hes in a plane, and (2) 

 for points A, B,C ow Si line 



AB + BC + CA = . . . . (6) 



From the first condition the distance must be of the form 



—^ _ Fx {A, B) 



where Fi and Fo are bilinear f mictions of A and B. Putting A, B 

 and C equal in the second condition we get 



^^1 = 0. 

 Hence 



F, {A, A) = . . . . (8) 



In this last equation replacing A by .1 + B and cancelling the terms 

 Fi {A, A) and F^ (B, B) we have 



F,{A,B) + F,{B,A) = 0. ... (9) 



The numerator oi AB must then change sign when we interchange 

 A and B. In (6) putting C = B we have 



AB-\- BA = 0. 

 This shows that 



F2 {A, B) = F. {B, A) . . . . (10) 



or the denominator of AB is symmetric in A and B. Let C = A -{- B. 

 Then (6) becomes 



F, (A, B) F, {B, A) + F, {B, B) F, {A, A) + F, {B, A) _ 

 F, {A, B) "^ F, {B, A) + F2 {B, B) ^ F, {A, A) + F^ {B, A) 



Making use of (8), (9) and (10) this becomes 



F, {A, B) [F2 U, A) Fo {B, B) - F^ {A, BY] = 0. 



