Quaternions to the Theory of Belativity. 44 5 



8. The equation of motion is to be written 



where P = F + A and the relation between the scalar and 

 vector parts o£ P is 



£KA=S~F, 



dr 



and this is the same as 



sk™p=o. 



9. An examination of P shows that it is constructed 

 so that 



F=£p and A = ik-y-, 



where p is the force as it enters into ordinary mechanics, 



and -=- is the rate of change of energy. 



We may easily derive the force in S' in terms of the 

 S measure. We have merely to transform P', 



P'=F'+A' = Q(F + A)Q. 



Equations (i.) and (ii.) give immediately 



F' = A(l-£ 2 )u + F + v(l-/8)SFv . . (xiv.) 

 and A' = £A + (l-£ 2 )*SFv .... (xv.) 



at 

 Using the ratio p given by (v.) 



p-rv(l-/5)Spv-^v^ 

 i at / • \ 



P = 571 o a > • • ( XV1 -) 



and in the same way 



dio ~ 

 dt' " 1 + vSuv * 



(xvii.) 



These two equations represent the general transformation, 

 and there is no particular direction for the vectors occurring 

 in them. Equations (xi.) are particular cases. 



As an example, we may apply the transformation to the 

 mechanical force on a moving charge. 



