( 441 ) 



As to the corresponding vibratory motions, we shall require that 

 at corresponding times, i. e-. for equal values of e", the configuration 

 of .S may always be got from that of S^ by the above mentioned 

 dilatations. Then, it appears from (7) that a"x, a"^, a", will be, 

 in both systems, the same functions of .i",y",z",t', whence we con- 

 clude that the equations (le) — (IVe) can be satisfied by values of 

 ^"r, .O''^, etc., which are likewise, in Sq and in «S, the same functions 

 of x", y", z", t". 



Always provided that we start from a vibratory motion in 5,, that 

 can really exist, we have now arrived at a motion in ;S, tliat is 

 possible in so fai' as it satisfies the electromagnetic equations. The 

 last stage of our reasoning will be to attend to the molecular forces. 

 In S^ we imagine again, around one of the ions, the same small 

 space, we have considered in § 7 and to which the molecular forces 

 acting on the ion are confined; in the other system we shall now 

 conceive the corresponding small space, i.e. the space that may be 

 derived from the first one by applying to it the dilatations (6). As 

 before, we shall suppose these spaces to be so small that in the 

 second of them there is no necessity to distinguish the local times 

 in its different parts; then we may say that in the two spaces 

 there will bo, at corresponding times, corresponding distributions of 

 matter. 



We have already seen that, in the states of equilibrium, the 

 electric forces parallel to O.Y, OF, OZ^ existing in S differ from 

 the corresponding forces in S^ by the factors 



1 1 



, — and — ,. 



From (Vg) it appears that the same factors come into play when 

 we consider the part of the electric forces that is due to the vibra- 

 tions. If, now, we suppose that the molecular forces are modified 

 ill (juite the same way in consequence of the translation, we may 

 apply the just mentioned factors to the components of the <o^«/ force 

 acting on an ion. Tiien, the imagined motion in S will be a pos- 

 sible one, provided that these same factors to which we have been 

 led in examining the forces present themselves again, when we 

 treat of the product of the masses and the accelerations. 



According to our suppositious, the accelerations in the directions 



of OX, OY, OZ in S are resp. — , -~~ and — — times what they 



30 



Proceediugs Royal Acad. Amsterdam. Vol. I. 



