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C The hyperbolic Spn- 

 I. Let us suppose a system of rectangular coordinates, so that 



ds =1 \/A^U^^ -\- . . . . AnUn^, 



and let us suppose a linevector distribution X with components 

 Xi,,. Xn, then the integral of X along a closed curve is equal to 

 that of a planivector Y over an arbitrary surface bounded by it, 

 in which the components of Y are determined by : 



Y is the fi}\bi derivative or rotation of JT. 



Further the starting vector current of X over a closed curved 

 Spn—\ is equal to the integral of the scalar Z over the bounded 

 volume of that Spn—i ; here 



z — — — > 



A^ An ^^ ^X^j.^ 



Z is the second derivative or divergency of X 



II. If X is to be a oA", i.e. a second derivative of a planivector 



S", we must have: 



^=<i = -I r~ 2^ 



x. =^ . - . 7. '^ ^ 



?1 



The necessary and sufficient condition for this is: 



If X is to be a o A, i. e. a first derivative of a scalar ff, we must 

 have: 



X — ^^ 



The necessary and sufficient condition for this is: 



r=o. 



III. Tlie oX, which has as divergency an isolated scalar value in 

 tlie origin (comp. Opitz., Diss. Göttingen, 1881), is directed along 

 the radius vector, and if we put the space constant equal to 1 is 

 equal to 



1 



It is the first derivative of a scalar distribution 



