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Mathematics. — ''The force field of the non-Euclidean spaces 

 with positive curvature" by Mr. L. E. J. Brouwer. (Commu- 

 nicated by Prof. D. J. Korteweg). 



(Communicated in the meeting of September 29, 1906). 



D^). The spherical Sp^. 



I. The theorems under C ^ I and II hold invariably for the sphe- 

 rical and elliptical *S/>„'s. But on account of the fmiteness of these 

 spaces we need not postulate a limiting field property for the 

 following developments. We shall first consider the spherical spaces. 



Firstly we remark for the general linevector distribution of the 

 spherical Spn that the total sum of the divergency is 0; for the 

 outgoing vectorcurrents out of the different space-elements destroy 



each other. This proves already that as elementary qX we can but 

 take the field of a double point. 



Schering (Göttinger ^Nachrichten 1873), and Killing (Crelle's Journal, 

 1885) give as elementary gradient field the derivative of the potential 



-; — = Vn (r). ') 



stn"~^ r 

 r 



But the derivative of this field consists of two equal and opposite 

 divergencies in two opposite points; and it is clear that an arbitrary 

 integral of such fields always keeps equal and opposite divergencies 

 in the opposite points, so it cannot furnish the general divergency- 

 distribution limited only to a total divergency sum =0. 



II. If we apply for a spherical Spn the theorem of Green to the 

 whole space (i. e. to the two halves, in which it is divided by an 

 arbitrary closed Spn-\, together), doing this particularly for a scalar 

 function <f which we presuppose to have nowhere divergency and 

 a scalar function having only in t^^'0 arbitrary points P^ and P^ 

 equal and opposite divergencies and nowhere else (such functions 

 we shall deduce in the following), we then find 



i. 0. w. <p is a constant, the points P^ and P^ being taken arbitrarily. 



1) A, B and G refer to : "The force field of the non-Euclidean spaces with nega- 

 tive curvature". (See these Proceedings, June 30, 1906). 



2) We put the space constant = 1, just as we did in the hyperbolic spaces. 



