220 PROCEEDINGS OF THE AMERICAN ACADEMY. 



where / and g are both arbitrary functions of -. . 



x L 



By this transformation a curve of the form, 



«G + sH= 1, 



will be transformed into a straight line and the theory of distance will 

 be the same if these curves were used for straight lines. There is 

 then an infinite group of transformations of the plane into itself which 

 will preserve distance and consequently will preserve angle. The 

 foregoing is then only one of an infinite number of geometries which 

 can be built up and which are simply isomorphic with the one here 

 discussed. 



Application to Minimum Developables. 

 The equations of a minimum developable can be written in the form, 



-»© + *•©• 



y = S (£\ + V*S' fU 



( f\ + v*T g 



where R' 1 + S' 2 + T' 2 = 0. 



u 

 Primes denote differentiation with respect to . Then 



v 



ds= ^R" 2 +S" 2 +T" 2 (udv-vdu). 



By a proper choice of variable the expression under the radical can 

 be made equal to unity. In fact the change of variable is, 



. r d (~ 



U 1 I \V 



wi r 



v l= ~~ J (R" 



(R" 2 + S" 2 +T" 2 )*- 



With this choice of variable we have, 



ds = udt — vdu. 



Since this form is the same for all minimum developables it follows 

 that all such surfaces can be mapped on any one of them in such a 



