Further Contributions to Non-Metrical Vector Algebra. 115 



of R being a pure convention. If, however, the equations 

 of motion can be put in the canonical form, there is a definite 

 reason of convenience for taking R=l; for this is the only 

 value that makes the " force" the gradient of a scalar 

 potential not involving the velocities explicitly. Thus the 

 force is defined as (jx,fy>fz) and the energy* as iyu-cV 2 - 

 The "mass" similarly becomes fxy 2 . 



4. In accordance with the theorem first quoted we have 



tfLdr=Q, .' • • (10) 



, T dx du , dz du TT 



-here L =p x ^ +Pl/ - +p. ^ + Plf % - H 



= U + constant. . .'". . . ... (11) 



Thus the only possible form of L is that obtained by 

 Silberstein, namely a constant + a scalar independent of the 

 velocities, which is itself the force-function from which the 

 force is derived. 



IX. Further Contributions to Non-Metrical Vector Algebra. 

 By L. Silberstein, Ph.D., Lecturer in Mathematical 

 Physics at the University of Mome'f. 



1. TN a recently published little book J I have en- 

 X deavoured to build up a simple vector algebra 

 independent of the parallel-axiom and of the axioms of con- 

 gruence, and therefore of any idea of measurement by means 

 of " rigid " bodies. This algebra consisted in essence only 

 of addition and equality of vectors. Notwithstanding this 

 poverty of means the said algebra was shown to be sufficient 

 for the treatment of any purely geometric (i. e. projective) 

 problems; for any such problem is ultimately concerned 

 with crosses of straights, joins of points, with concurrency, 

 collinearity, and similar questions, and all of these can be 

 covered by the sole operation of addition, with its inverse 

 and its iterations. No urgent need, therefore, was felt for 

 the introduction of a " multiplication " of vectors by vectors. 

 Also, at the time of writing and up to the last moment of 



* This possibility of this value of the energy is indicated by Dr. "Wilson 

 (loc. cit.). 



t Communicated by Prof. A. N. Whitehead, F.K.S. 



X ' Projective Vector Algebra,' au algebra of vectors independent oi 

 the axioms of congruence and of parallels. Pp. 78. Bell & Sons 

 London, 1919 (May). 



12 



