420 PROCEEDINGS OF THE AMERICAN ACADEMY. 



be discussed are so much more readily visualized in this simple case 

 that we shall consider a few important theorems before entering upon 

 the treatment of three and four dimensional manifolds. 



The meaning of the mass of a particle, when that mass is determined 

 by a person at rest relative to the particle, will be taken as understood. 

 We shall call that value of the mass viq. Let us consider a (5)-curve 

 which represents the locus in time and space of this material particle, 

 and at any point of the locus a tangent of unit interval (or unit tan- 

 gent) w. By multiplying w by the scalar mo, we make a new vector 

 which we shall call the extended momentum. If now we choose any 

 pair of axes z and t, the slope of the locus with respect to these axes, 

 that is, the velocity of the particle, we have called v. The momentum 

 vector may then be written, by (5), 



woW = , ki + . k4. (10) 



Vl — 1,2 Vl — 2,2 



If the /-axis were chosen parallel to the tangent w, the coefficient 

 of k^, that is, the component of the extended momentum moW along 

 the time axis, would be simply viq, the stationary mass. If, as we 

 have assumed, the particle is regarded as moving with the velocity 

 V, we shall take the component of ??ioW along the /-axis as the mass m. 

 In other words, the mass of a body appears to increase with its velocity 

 in the familiar ratio 



m = . (11) 



Vl — 1,2 



The component along the a'-axis is then mv, the momentum. We 

 may therefore write the vector of extended momentum as 



moW = mv]s.i + mlHi. (12) 



22. From our equation for the curvature we may write 



c^moW dinv. . dm. 1 f dmv, , dm, \ ,,-v 



The vector moC we shall call the extended force. Since our ordinary 

 definition of force is time-rate of change of momentum, it is evident 

 that the .r-component of the extended force multiplied by Vl — ^2 Jg 

 ordinary force. That is, 



/; o dmv , , 



; = Vl —v^moc^= ~^. (14) 



