PHILLIPS AND MOORE. — LINEAR DISTANCE AND ANGLE. 69 



where a represents the plane BCD, in which the base Hes and k 

 is a function of A and o-- By a series of motions consisting of moving 

 one triangle of the tetrahedron in its plane it is seen that the tetra- 

 hedron can be moved into any other having equal volume. These 

 motions keep the volume constant and therefore k is an absolute 

 constant. Hence choosing our unit so that k = 1, we have 



{A BCD) 



'^^^^- (ct>A){<t^B){4>C){4>D) ' ' ^'^^ 



From the definition we have 



ABCD = BCD-Aa = BCDA{BCD) 



\ BC+CD+DB 



= AB-CD + ACDB + AD-BC . . . (25) 



From (24) we see that if the vertex A lies in the plane ABC the 

 volume is zero. Hence applying this to (25) we have 



'ab'cd + IcWb-^ ad-Jc = 



as a relation connecting four points lying in a plane. This relation 

 is seen to be identical with the relation connecting the Pliicker co- 

 ordinates of a line. From this a theory of plane c^uadrilaterals could 

 be built up. 



20. Summary. We have defined a bilinear function of any two 

 spaces in three dimensions. In case one of these spaces is a point 

 we call this function a distance otherwise an angle. We have also 

 defined certain areas determined by three elements and volumes 

 determined by four. These functions are all invariant under a six 

 parametered group of coUineations projectively equi^'alent to the 

 group of coUineations leaving euclidean volume invariant. Under 

 the correlation 



.T y = const. 



each of these functions is equal to the dual function of the transformed 

 elements. The expressions for these functions are 



-—__U^AB1_ (,,. 



''''- {^A)i^B) ■ • ■ ■ (11) 



—^^Jp^R^ .... (12) 

 ' {Fa){F^) ^ ^ 



