MISCELLANEOUS PAPEKS. 229 



as it also appears from the figure. The deduction of formulas (1) 

 from an actual perspective construction has the great advantage that 

 all results gained from the analytical discussion may easily be inter- 

 preted constructively and geometrically. It is noticed that a per- 

 spective transformation depends upon three essential constants, i. e., 

 if its center is fixed. This is equivalent to saying that the axis of 

 perspective can be chosen in a doubly infinite number of ways, and 

 the constant k in a singly infinite number. If to the perspective 

 transformation P, as given by (1), we apply a dilation D, defined by 



x"=dx', y"=y', (2) 



and which may be considered as a special case of perspective where 

 the center is infinite in a direction perpendicular to s. From this it 

 is seen that the combination of a perspective and dilation ( PD ) may 

 be expressed by 



(3) 



Applying to this transformation consecutively a transformation by 

 equal areas (A), defined by 



x"'=x" + ky" ) 



y"'=y" ) 



and which may also be considered as a special case of perspective, in 

 which C is infinitely distant in a plane perpendicular to the bisecting 

 plane of II and II' ; then a transformation (T), defined by 



(4) 



xW = x'" 4- V 



[ , (5) 



and finally a rotation ( R 



x<5)=nxW — my(4) 



y(5)=nxW+my(*) ^ ' ^ ' 



(m2 + n'-* = l), 



we arrive at a transformation of the form 



ax + by + c ~] 



y 0) 



X — 



dx -t- ey + f 



> _ gx + by -f 3 , 



y dx + ey + f ■ J 

 This transformation is characterized by eight independent con- 

 stants, and is called, as is well known, a projective transformation, or 

 coUineation. It may be considered as the result of the combined 



transformations 



(PDATR), (8) 



which are determined by 3, 1, 1, 2, 1 parameters, respectively. It is 



