414 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



_ clw _ d-r 

 ds ds- 



is the curvature, taken as a vector normal to the curve. 



vk4 n dv 

 dzi. 



Hence 



r ki vk4 n dv 



L(i — t"? (1 — 1)-)2 J dx 



(8) 



(9) 



In magnitude the curvature is 



dv 

 dxi 



V 



c«c = 



d^X\ 

 dzi 



(1 - .^)^ 



1 — 



dxj 



Relative to axes k/, k4', the result is 



c = 



k/ 



+ 



v"^ 



(1 — v'^f ' (1 — v"'-Y_ 



dv' 

 d^' 



(1 — uv) ki' 



_(1 — v-'Y Vl 



+ 



U" 



(v — u) ^' 



(1 — v'-y vr= 



w 



In complete analogy with the circle in Euclidean geometry the 

 pseudo-circle in our non-Euclidean geometry has a curvature of con- 

 stant magnitude throughout. The curvature of any other curve may 

 always be represented as the curvature of the osculating pseudo-circle, 

 and in magnitude is inversely proportional to the radius of that pseudo- 

 circle. 



Kinematics in a Single Straight Line. 



18. Before proceeding to the discussion of the non-Euclidean geom- 

 etry of more than two dimensions we may consider some simple but 

 fundamental problems of physics which may be treated with the aid 

 of the results which we have already obtained. 



The science of kinematics involves a four dimensional manifold, 

 of which three of the dimensions are those of space, and one that of 

 time. By neglecting two of the spacial dimensions, in other words 

 by restricting our considerations to the motion of a particle ^^ in a 

 single straight line, kinematics becomes merely a two dimensional 

 science. The theorems of kinematics, not in the classical form, but in 

 the form given to them by the principle of relativity, are simply 

 theorems in our non-Euclidean geometry. 



19 By particle we do not as yet mean a material particle but merely an 

 identifiable point in motion. 



