46 



KANSAS UNIVERSITY QUARTERLY, 



being transformed to P^. The cross-ratio of the pencil through 

 the vertex C is 



_ sin ACP sin ACP,_Pb 



k,-^C(ABPPj— sin BCP ' sin BCP " 



Pa 



-^^ ^. But the 

 P. a. 



perpendiculars from P and P, on the sides of the triangle are 



proportional to the areas of the triangles of which they are the 



, Pb .P,b,_APAC. APaAC 

 Hence ky= 



altitudes. 



Pa 



P,a, 



,PBC APiBC 



.\PAB APjAB ,, AtPBC APiBC 



In 1 ke manner kv=-~^ : ^=^^^ — ; and k, — ^ : -r-z — i . 



in like manner Kx ^ p^c APjAC • — APAB APiAB 



We easily verify that kxkyl</,= i. But P and P, were taken to be 



any pair of corresponding points in the plane. Remembering the 



theorem* that the cross-ratio of the invariant elements and any 



pair of corresponding elements in a one-dimensional projective 



transformation is constant for all pairs of corresponding elements, 



we have found the following important theorem: 



77/rorr/// 2. — llic cross-ratio of flw arras oj Jour triaiii^lcs ic/iosc 



vertices arc any pair of corrcspo/idi/ii:^ points in tlic transfor)nation T 



and whose bases arc any /vfc sit/i's op tlie invariant triangle op T is 



constant for ait pairs op cor responJing points. 



5. Implicit Normal Form of Equation of Type I. The equations 

 of a projective transformation are usually given in the form 



_^_ax + by + c_ ^^^ a,x+b,y+c^ ^^^ 



^ a2xAb._,yAc„ ' aoX-f-bgyAc^, 



When these equations represent a transformation T, they can be 

 thrown into a normal form in which the constants in the equation 

 are the coordinates of the three invariant points and the character- 

 istic cross-ratios along the invariant lines. In Fig. 4 let the 



*K. U. Quarterly, Vol. 1v, 18!)."), p.,;i. 



