214 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



Curvature. 

 then have 



Here as in case I we can define curvature as 



da 



ds 



We 



-r- _ da dy d 2 x dx d 2 y 

 ds ds ds 2 ds ds 2 



(8) 



dx 

 dy\ 2 d^ (lte_ 

 ds J ds \dy 



ds 



which is the same form as the formula for curvature in euclidean geom- 

 etry. From (8) we see that curves of zero curvature are straight lines. 

 If x is taken as the independent variable the differential equation of 

 the curves of constant curvature is, 



d 2 y ( dy ^ 



the solution of which is, 



ay — c 2 x- = V ( Cl 



or 



Kx 2 ) 

 (ay — ax) 2 + K.r 2 = a. 



We will call these curves of constant curvature "pseudo circles." 

 From the equations it is seen that these are conies with respect to 

 which / is the polar of F. The points of a curve for which K = 0, 

 are points of inflection. Any collineation of the plane which leave / 

 and F fixed have K = 0, as an invariant. 

 Any conic of the form, 



Ai.r 2 + 2A«xy + A 3 y 2 = 1, 

 is a pseudo circle whose curvature is, 



K = \ (AxA 3 - A 2 2 ) 



The line area of a triangle is the sum of the sides. As in Case I, 

 this can be expressed as, 



