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me to relate tlie rotation in S^ to two rotations in aSV The relations 

 between the elements of the fonr-dimensional rotation and the elements 

 of the two threedimensional rotations belonging to it have been 

 explicitly pointed out bv me in "Sitzungsberichte der Berliner 

 Akademie" of Jnly SO^'i 1896 and in the "Journal fur die reine 

 und angewandte Mathematik" Vol. 118, p. 215, 1897. I have 

 particularly found that the components of the velocity of the first 

 rotatio]! are easily deduced from the components of the velocity of 

 the two others (compai'e also my lecture at the "Naturforscher Ver- 

 samndiing" at Hamburg 1901 : "On rotations in fourdimensional 

 spaces," (Ueber Drehungen im vierdimensionalen Raum). 



Mr. Brouwer arrives in his paper also at these results though in 

 a ditferent way, namely geometrically, whilst I have worked alge- 

 braically. Mr. Brouwer arrives at a decomposition ("Zerlegung") of 

 the fourdimensional rotation into two threedimensional ones, whilst 

 I use the expression coordination ("Zuordnung"). 



Berlin, March 28^'>, 1904. 



Mathematics. — "A/</ebraic deduction of the decomposabilitij of 

 the continuoas ^notion about a fixed point of S^ into those of 

 two Si's". By Mr. L. E. J. Brouwer. (Communicated by 

 Prof. Korteweg), 



As the position of aS'^ is determined with respect to a fixed system 

 of axes by six independent variables and that of 6', with respect to 

 a fixed system of axes by three independent variables we understand 

 at once that in an infinite number of ways two S^ motions can be 

 coordinated to an S^ motion, so that position and velocities ofxS^are 

 determined by position and velocities of the two >SVs. On such a 

 coordination Mr. Jahnke has been engaged in the papers mentioned 

 above and has deduced the relations between positions and velocities 

 of S^ and the two aSVs. Interpreted geometrically his coordination 

 amounts to tlie following : Let us suppose in ^S'^ a fixed system of 

 axes A\ A^^ A', A^^, and a moxable one Y^ Y.^ Y^ Y^; let us con- 

 sider the part equiangular to the right of the double rotation, which 

 transfers A\ A^., A"^, A'^ into Y^ Y^ Y^ Y^ ; let us add to it an equal 

 equiajigular double rotation to the left (namely equal with respect to 

 the system of axes A\ A^ A^, A^^ ; only with respect to a definite 

 system of axes can we call an ecpiiaiigular double rotation to the 

 right and one to the left equal) ; the resulting rotation becomes a single 



