( 66 ) 



Mathematics. — '* Polydimensional Vectordistributions" . ^) By L. E. J. 

 Brouwer. (Communicated by Prof. D. J. Korteweg.) 



Let us call the plane space in which to operate >S„ ; we suppose 

 in it a rectangular system of coordinates in which a 6^ represents 

 a coordinatespace of p dimensions. Let a /^^-distribution be given 

 in Sn', ie. let in each point of Sn a /9-dimensional system of vectors 

 be given. By Xa, «.^....a we understand the vector component parallel 



to C^t indicated by the indices, whilst as positive sense is assumed 

 the one corresponding to the indicatrix indicated by the sequence 

 of the indices. By interchanging two of the indices the sense of the 

 indicatrix changes, hence the sign of the vectorcomponent. 



Theorem 1. The integral of f'X in S» over an arbitrary curved 

 bilateral closed S^ is equal to the integral of ^'+^ F over an arbitrary 

 curved *S^+i, enclosed by S^ as a boundary, in which P + ^l^is 

 determined by 



n. 



XX. 



^ 



ÖX. 



^2 78 



V+1 



d.v. 



where for each of tlie terms of the second member the indicatrix 

 {ctfj^aq^. . . aq Oq ) has the same sense as {a^ «, . . . «^-|-i). We call 



the vector Y the first derivative of )'X. 



Proof. We suppose the limited space Sp-\.\ to be provided with 

 curvilinear coordinates ii.^ . . . ?^y,-}-i determined as intersection of curved 

 C)/s, i. e. curved coordinatespaces of ;>-dimensions. We suppose the 

 system of curvilinear coordinates to be inside the boundary without 

 singularities and the boundary with respect to those coordinates to 

 be everywhere convex. 



The integral element oï f-^^Y becomes when expressed in differen- 

 tial quotients of rX : 



ÖX. 



Y. 



^,+x 



dx. 



Ya-i ... a 





"/>-l-l 



dxo 



^+^ 



du^^ 



d.v. 



du. 



du 



Z'+l 



6?Mj 



du 



/;+!• 



1) The Dutch original contains a few errors (see Erratum at the end of Ver- 

 slagen 31 Juni 1906), wliich have been rectified in this translation. 



