2o8 KANSAS UNIVERSITY QUARTERLY. 



The combined effect of a one-termed group of e /at tons and a certain 

 involution is tJic system of all iin'o/t/tions whose axes intersect eacJi other 

 in one and the same point. This point is determined by the intersection 

 of the axes of the original iiroo/i/tion and t/ir one-termed group of 

 elations. 



That really all involutions of the system are obtained is seen 

 from the trigonometric tangent of the resulting axis of involution, 

 which follows from (4), 



bja-|-a|b 



c^a + ajC 



In this expression a, b, c, bj, c^, are fixed quantities; but it may 

 assume any real value if aj varies, 



We have now discussed the the essential parts of our problem 

 and without entering into further details we will finally treat two 

 important special cases. 



2 SPECIAL CASES. 



(«) Orthogonal and oblique, or axial sxmnwtrx, may be considered 

 either as a linear, or as a special perspective involution in which 

 the center is at infinity. The analytic ' representation will be 

 identical in both cases if we choose the origin in the axis of 

 symmetry, and let this be the X-axis, and let the Y-axis be parallel 

 to the axis of elation. The center of the perspective involution 

 and of the one-termed group of elations is the infinitely distant 

 point of the Y-axis. Using either orthogonal, or oblique coordi- 

 nates the equations of the axial symmetry may be written 



and those of the elation 



Xj=X, 



Xg — Xj, 



y2=mx,+y,. 

 Applying the elation to the symmetry the result is the involution 

 and especially the axial symmetry Xg=x, 



yg^mx- y, 

 having as its axis 



m 



or in words: 



The effect of a one-termed gri>/f/' of linear elatio)is upon a certain 

 axial symmetry liai'ing the same infinite center is the system op all axial 



