442 Mr. H. T. Flint on the Applications of 



Similarly, we may derive 



11 V 1 



and u z 



y 0(1 -vu x ) "* 0(1 -vu x ) 



by taking v along Oy and Oz respectively. 



3. Let m denote the " rest mass " of a particle, and then 



Jsl = m io 



is also a physical quaternion — the momentum quaternion. 

 We obtain from § 2 



m m 



0(1- vu x ), . . . (vii.) 



(l-u'*)i (1-U 2 )i 



or writing this 



m' = m0(l — vu x ) 



we obtain the usual transformation for mass. 



4. By a second differentiation we pass to acceleration and 

 write : 



- dw 



Transformation of acceleration is completely expressed by 



/'=Q/Q. 



5. We write P = w / 



and call P the physical force quaternion. 

 We then have 



(cW dH 2 \ +>, ~ (dH d 2 l\„ 



Thus 



d 2 v' dH d 2 v d 2 v 



m °dr2- =imol3v7 ^ +™ Q — 2 + v(l-£)ni S^-j . v, (viii.) 



and 



d 2 l' Q <Pl _ , Q „d 2 r - s 



m oTT =m oP J >> +niQipvb -T-f. v (ix.) 



If v is along Ox we find from (ix.) 

 dm' dm mv du x 



dt' dt (1 — vi( x ) ' dt 



(X.) 



