218 PROCEEDINGS OF THE AMERICAN ACADEMY. 



(a — ky y )x + (b + kxi)y = ax\ -f- by h 

 where 



will be the e volute. For the pseudo circle this is again a pseudo 

 circle. In fact if the curve is of the form f(x, y) = 1, where f(x, y) 

 is a homogeneous function of x and ?/, the order of the evolute will be 

 the same as the order of the original curve. The dual of this set of 

 curves is the set traced by the points on the tangent line at a fixed 

 distance from the point of tangency. In the case of a pseudo circle 

 these curves are again pseudo circles. 



The pedal curves of a given curve form another interesting configura- 

 tion. Here for a given point there will be an infinite number of pedals 

 depending on the size of the angle used. If we denote the given curve 

 by C and the dual of the parallel by P and take A as the point with 

 reference to which the pedal is taken a simple construction will show 

 that the pedal curve is the locus of the intersections of the tangents to 

 C with the lines drawn from the corresponding points of P through A. 

 The point A will be a multiple point on the pedal of multiplicity equal 

 to the class of C. 



Length preserving transformations of the plane into itself. We found 

 that there was a three papameter group of collineations which left 

 distance invariant. It is evident that there are other transformations 

 which leave distance invariant since the distance between two points 

 measured along various curves may be the same as if measured along 

 straight lines. The transformation, 



(T) a- = G(x\ if), y = R(x\ y l ), 



will preserve length if, 



= x 1 dy 1 — y 1 dx 1 

 from which we see that, 



