GEOMETRY WHOSE ELEMENT OF ARC IS A LINEAR 



DIFFERENTIAL FORM, WITH APPLICATION TO THE 



STUDY OF MINIMUM DEVELOPABLES. 



By C. L. E. Moore. 

 Presented, March 10, 1915. Received, February 24, 1915. 



We are accustomed to think of the element of arc as being defined 

 by the square root of a quadratic differential form. This defines 

 length as a pure number which is always taken as positive. Under 

 this definition the distance between two points cannot have a sign 

 attached unless it is fixed arbitrarily for each direction of the surface. 

 There are however two well known classes of surfaces for which the 

 element of arc is a linear differential form, viz., minimum planes and 

 minimum developables. For the minimum plane the element of arc 

 is an exact differential, while for the minimum developable the element 

 of arc is not an exact differential. These are the only two possibilities 

 for a linear differential. By a proper choice of variable one exact 

 (inexact) differential form can be transformed into any other exact 

 (inexact) form but a form of one kind cannot be transformed into the 

 other kind. The linear form of the arc length shows us first of all. 

 that on both these surfaces distance is a directed quantity since the 

 sign depends on the direction of integration. On a. minimum plane 

 the length of any closed curve is zero while on the minimum develop- 

 able this is not necessarily true. These surfaces then become all the 

 more interesting because they differ in such a marked respect from 

 ordinary surfaces. 



The element of arc being linear, it is not possible on either of these 

 surfaces to define geodesies as the curves which minimize the integral, 



f&dz + Ydy), 



for such an integral, by varying the path can take any value whatever. 

 For the study of the geometry on such a surface then it is necessary 

 to find some other means of defining what corresponds to straight lines 

 in the plane. As for the minimum plane, straight lines are already 

 defined by means of linear equations in the parameters ordinarily 

 used or by the intersection of the minimum plane with an ordinary 

 plane. However the ordinary definition of angle breaks down so 



