290 M. L. Lorenz on the Identity of the 

 let a new function X2 be denned, in which by the relations 

 e'f t ) and e ! lt J, where a is a constant, it shall be ex- 

 pressed that these are the same functions of It ) as e' and 



e' are of t in the above expression H. Now by the development 



of the series we have 



-<J 



dt a^ dt 2 a 2 1.2 



/ r\_ ,_dJ r_ dV r 2 _ 1 



(,r\_ ,_<& r dV r 2 1 

 V a)~ e dt *« + dt 2 'a 2 'l. 



which series are inserted in the above equation, and this is then 

 differentiated with respect to x. There is thus obtained 



de 1 de 1 



and if in this equation for -=- and -j the values given by equa- 

 tions (2) are substituted, we obtain by partial integration 



dU_da 1 d CCCdx'dy'dz' ,,l_dV_ 



dx~~dx a 2 dijjj r U+ a* dt '"> W 



U having here its previous meaning. Hence we may put 



da.id ccr &WdJ ./ r\_da i dv 



^ « 2 ^JJJ J U V a)-dx + a* dt' ' W 

 where it is indicated by u'lt — -) thatw' is here a function of 



( / j instead of t alone. 



The right hand of this last equation is a series of which only 



the first two members are retained, and whose following mem- 



v c 



bers proceed by increasing powers of -. If a be assumed = -, 



both these members become the same as the expression between 

 the parentheses in the first of equation (4); but now, according 

 to Weber's determination, 



c= 284736 miles, 

 while the greatest value of r in the experiments has only exceeded 



