CONTINUOUS GROUPS OF PROJECTIVE TRANSFORMA'ITONS. 



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may be different. Thus special cases arise when the conies touch 

 one another, or touch the lines 1 at the same point, or have con- 

 tact of the second or third order, etc. These special cases we now 

 proceed to examine. 



Let us first consider the case of a projective transformation where 

 the two conies touch one another. The invariant figure in this case 

 consists of y, the common tangent to the two conies at A their 

 point of contact, and another common tangent x intersecting y at 

 B, (Fig, 2 (a) ). It does not matter whether the two conies inter- 

 sect in two other real or imaginary points, the invariant figure is 

 the same in either case. This kind of a projective transformation 

 may be considered as a special case of the last when two sides of 

 the invariant triangle coincide. We shall call this Case II. 



Instead of simple contact as in last case the two conies K and K^ 

 may have contact of the second order at a point A. When the 

 conies have a contact of the second order they can and must inter- 

 sect in one other point. In this case the common tangent to the 

 two conies at A is the only invariant line of the transformation and 

 the point A is the only invariant point on the invariant line, (Fig. 

 2. (b) ). We shall call this Case III. 



Again the two conies may both touch the line 1 at the same point, 

 the contact being of the first order. In this case the two conies K 

 and Ri have two other common tangents x and y which intersect 

 at some point C. (Fig. 2 (c) ). It is at once evident from the figure 

 that the transformation determined by K and K^ leaves the lines x 

 and y and the point C invariant. A little further consideration of 

 the construction shows that the points A, B, and L on the line 1 are 



