WILSON AND LEWIS. — RELATIVITY. 14." 



poneiits of the corresponding extended vectors. Indeed the space 

 component of the extended velocity is V/Vl — ti-. The ordinary 

 force, measured as rate of change of momentum, is 



dv du 



dm\ ay , d)n dt , dt ,,_, 



'= dt ='%, + '* =^i_7)? + (r~7)V <*^> 



which is the space component of tmqC multiplied hy Vl — ji-. 

 It is evident that in our mechanics the equations 



f = — ,- and f = m a, 

 dt 



where a = dv/dt, are not equivalent, and it is the first of these which 

 we have chosen as fundamental. This makes the mass a definite 

 scalar property of the system. Those who have used the second of 

 the equations have been led to the idea of a mass which is different 

 in diti'erent directions, and indeed have introduced as the " longitudi- 

 nal" and the "transverse" mass the coefficients 



mo mo 



(1— «2)t (1—^,2)2 



of the components of acceleration along the path and perpendicular 

 to it, that is, of the longitudinal and] transverse accelerations, which 

 are respectively 



dv du 



The disadvantages of this latter system are obvious. 



An interesting case of planar motion is that under a force constant 

 in magnitude and in direction, say fx = 0, fy = — k. The momen- 

 tum in the a:-direction is constant, that in the ^/-direction is equal to 

 its initial value less Jet. From these two equations the integration may 

 be completed. Or, in place of the second, the fact that the increase 

 in mass (that is, energy) is equal to the work done by the force, may be 

 used to give a second equation. The trajectory of the particle is 

 not a parabola, but a curve of the form y -\- a = — b cosh (ex — d), 

 resembling a catenary. 



The space-time locus of uniform circular motion is a helix 



r = a {ki cos nt + k2sin7iO + k^^- 



