444 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



To obtain the vector curvature of the locus we write 

 (hf dxid-Vf 1 



c = 



ds 



or 



ds dxi 

 1 



f/v V + k4 dv 



i—v" dxi "^ (1— «2)2 ^ ^; 



_ f/v V + k4 dfo 



^ ~ r=^2 ji + (1 _^2)2 ^ 7^- 



(43) 



If V be written as V = vn, where u is a unit vector, the resolution of c 

 into three mutually perpendicular components along u, k4, and dM 

 follows immediately: 



u 



c = 



dv 

 dt 



+ 



du 



vk, 



(i — v'-y ' 1 



The magnitude of C is 



'dvV 



Vc»c = 



+ 



dv 

 dt 



1 



(1 —2)2)2 



„c?U du~ 

 dt dt 



L(i — v'-y ' (1— «2)2_ 

 1 



(44) 



dt 



+ 



(45) 



v«v + 



1 — V' 



(vv) 



a[v«V — (vxv)•(vxv)]^ 



1— «2 



1^ 



In case the acceleration is along the line of motion, these expressions 

 reduce to those previously found; the additional term is due to 

 the acceleration normal to the line of motion. 



36. Mass may now be introduced just as in the simpler case already 

 discussed, and here likewise we are led to the equation 



TO = 



mo 



Vl — t>2 



The extended momentum in this case is also TOqW, that is, 



WoW = mv + mki. (46) 



We may speak of W as the extended velocity, of c as the extended 

 acceleration, and of trioC as the extended force. It is to be noted that 

 while ordinary momentum is the space component of extended momen- 

 tum, ordinary velocity, acceleration, and force are not the space com- 



