PHILLIPS AND MOORE. — LINEAR DISTANCE AND ANGLE. 59 



We choose the unit angle such that ^' = 1 . Then 



F=[ck p]. 

 also ^^ 



[F p] = [(f> p. ;j] = I (p. p)(f) = (f) 

 if we choose the magnitude of p such that 



ipp) = 2. 



The relations between ^ and F are then symmetrical. 

 Our formulae are now 



(pAB ) 



^^ {4>A){cf>B) ' • • • ^^ 



— _ (p-al3) 



"^~(Fa)(F/3) • • • • U^> 



with the condition that F = [4> p], (f> = [F p] and (/>p) = 2. 



The ratio of two distances or of two angles, also the product of a 

 distance and angle are invariant under the seven parameter group 

 of coUineations leaving the complex p and the plane <^ fixed. If one 

 of these transformations leaves a distance or angle unchanged it 

 leaves all distances and angles unchanged. Those quantities then 

 are invariant under a six parameter group. Any tetrahedron can 

 therefore be transformed into an equal tetrahedron (one having equal 

 length of sides) by a collineation leaving distance and angle invariant. 



From the formula for the distance between two points, it is seen 

 that distances along a line of the complex p are zero provided neither 

 of the points lies on </). The distance along a line of p to ^ is inde- 

 terminate but along any other line it is infinite. Similarly the angle 

 between two planes intersecting in a line of p but neither passing 

 through F is zero. If one of the planes passes through F, the angle 

 is indeterminate or infinite according as the other plane does or does 

 not cut it in a line of p. 



11. The locus of points ?/ at a distance ^ from the point x is a plane 



11 The formula [^p-p] = | (p p) ^ can be proved as follows. Let 

 p = aP + y8. 

 Then ipp) =2(ap7 8j 

 and [<j>P-p] = [<t>(ap + 78). (ap + 78)] 



= (<j) a p 8) 7 — (<}) a P V) 8 + (<}) a 8 Pja — (<1> 7 8 a) p 

 = <}> (a p 7 8) = \{p p) ^. 



