Ball — On the Character of the Linear Transformation. 533 



The demonstration is as follows : — 



If the four double points of the homographic systems be taken as 

 points of reference ; then 



Vi = hi Xi, 



y% = K X3, 



The requisite condition is obviously 



hi hi = hi hi, 

 for then the quadric surface 



will merely transform to the identical surface 



^1^2 = Sy^y^. 



The condition admits of being stated geometrically. Each edge of 

 the tetrahedron of reference possesses the property, that correspondLug 

 points thereon form two homographic systems, of which the two 

 comers are the double points. Thus two corresponding points on 1 

 and 3 form with the corners the constant anharmonic ratio 



hi 



The two other comers of the tetrahedron define the double points 

 of the systems whereof the two components make with the comers the 

 constant ratio 



hz 



but since hi hi = A3 hx, these two anharmonic ratios are equal, and hence 

 we have the following theorem : — 



If the linear transformation represents displacements ; then two 

 opposite edges of the tetrahedron are such, that the anharmonic 

 ratio of two corresponding points, with the double points on one edge, 

 is equal to the corresponding anharmonic ratio on the other edge. 



_ This principle will prove the theorem which forms the subject of 

 this Paper. 



