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

 with negative curvature''\ Bj Mr. L. E. J. Brouwer. (Commu- 

 nicated by Prof'. D. J. Korteweg). 



A. The hyperbolic Sp^. 



I. Let us suppose a rectangular system of coordinates to be placed 

 thus that ds = \/ A^ du^ -\- B^ dv^ -f C^ dw^, and let us assume a line- 

 vector distribution X with components Xu, Xo, X^o, then the integral 

 of X along a closed curve is equal to that of the plani vector F over 

 an arbitrary surface bounded by it; here the components of Y are 

 determined by : 



_ 1 \ d{x,B) a (x^ c) \ 



For, if we assume on the bounded surface curvilinear coordinates 

 è and tj, with respect to which the boundary is convex, the surface 

 integral is 



r,^ /a. die dv dto\ f d {x„ B) a (x„, c)\ 

 J 2^\M ' d^~a^ • dïJV~a^;; aT";'^*'^'^- 



Joining in this relation the terms containing X^ C and adding and 



a {x,o c) dio aw 



subtracting — ^^ . ^;^ . v- we obtain : 



ƒ 



^^^ \h{X,oC) bw b{X^C) bw 

 di dy] 



a?j a§ a§ a?/ 



Integrating this partially, the first term with respect to % the second 

 to %, we shall get iXwCdw along the boundary, giving with the 



integrals j Xv B dv and | Xu A du analogous to them the line integral 



of X along the boundary. 



In accordance with the terminology given before (see Procee- 

 dings of this Meeting p. QQ — 78) ^) we call the planivector Y the 

 first derivative of X. 



1) The method given there derived from tlic indicatrix of a convex boundary 

 that for the bounded space by front-position of a point of tlie interior ; and the method 

 understood by the vector Xpqr... a vector with indicatrix opqr.... We can however 

 determine the indicatrix of the bounded space also by post-position of a point of the 

 interior with respect to the indicatrix of the boundary; and moreover assign to 

 tlie vector Xyqr... the indicatrix pqr...o. We then find : 



