PHILLIPS AND MOORE. — LINEAR DISTANCE AND ANGLE. 65 



is a necessary and sufficient condition that A, a be transformable 

 into A', a by a motion leaving distance invariant. Such a function 

 is 



(A a) 



Aa= .... (16) 



{<t> A) {F a) 



Let a be a plane BCD. Then 



{A BCD) 1 (.1 BCD) (p p) 



Aa 



(0 A) {F BCD) (cj) A) ict>p-BC D) 

 This expression can be written 



(p A B) {p CD) + {p A C) (pDB) + (p A D) (p B C) 



{4> A)\{4>B) {pC D) + {4>C) {pD B) + {cl>D) {pBC)\ 



ABCD + AC-DB + AD-BC 

 B~C+CD + Dli 



That ^ a is invariant under the transformations leaving distance 

 unchanged is shown by the last form. Conversely if 



A a = A' a' 



we take in a a triangle BCD and in a a corresponding triangle 

 B'C'D' such that 



A'B' = AB, A'C = AC, B'C = BC 



A'D' = AD> CD' = CD. 

 Then the above equation shows that 



B'D' = BD. 



The two tetrahedra have all their edges equal and hence the one is 

 transformable into the other. 



This quantity A a we call the distance from the point A to the 

 plane a. It has many of the properties of euclidean distance from 

 point to plane. Thus if the point lies in the plane (point not in (f> 

 and plane not through F) the distance is zero. If the plane is held 

 fixed and the distance kept constant the point lies in a plane cutting 

 a on (p. If the point is held fixed the locus of the plane is a point on 

 the line joining the given point to F. 



If - iT = X 



