( 833 ) 



rotation parallel to tlie s[)afe A\ A'^ A', which woiihl tranyfcr the 

 system of axes A^, A', A', into an othei' Z, Z^ Z,. Thus to each 

 position }" with respect to X^ X^ X^ X^ answers a position Z with 

 respect to A"i A', A"», and by interchanging right and left, in an 

 analogous manner a position U with respect to A^j A.^ X^ ; and we 

 may consider the positions Z and U as coordinated to the position Y. 

 Not immediately to be seen are the two following properties of 

 the S^ motion geometrically deduced in what was communicated in 

 the February meeting. 



1^^ The continuous motion of iS'^ can be decomposed, that is : 

 independent of the clioice of a system of axes two detinite three- 

 foldly infinite motion groups exist iji S^ in such a way that an 

 arbitrary motion can be composed out of two motions each of wiiich 

 belongs to one of the groups mentioned. 



2'«l The continuous motion of S^ can be decomposed into two 

 >Sj motions, that is, two t\vodimensional manifoldnesses (namely 

 those of the systems of planes equiangular to the right and to 

 the left) exist in S^ in such a ^vay that each of the motion 

 groups mentioned ti-ansforms the elements of one of them into 

 each other and leaves the other unchanged ; to which further- 

 more w^e can aHow^ two<limeusioual Euclidean stars to answer in 

 such a way that to congruent combinations in one of the manifold- 

 nesses congruent combinations of the Euclidean stars correspond, 

 that to the corresponding motion group oï S^ answers the motion group 

 of the Euclidean star movable as a solid and that to congruent 

 combinations in the motion group of S^ answer congruent combinations 

 in the motion grou]> of the Euclidean star movable as a solid : reason 

 wily we may call the two twodimensional manifoldnesses twodlmen- 

 slonal EucUdean stars and the motion groups of ^9^ transforming them 

 Euclidean threedimensional motion </j'onj)H ahout a lieed jioint. 



We shall now see how Ave can arrive algebraically at both results. 

 Mr. Jamnkk takes from Caspaky the so called "Elementary trans- 

 formation" (see a.o. Jahresl)ericht der Dentschen Mathematiker- 

 Vereiniging XI, 4, 1902, p. J80 and F. (Vvspary, Znr Theorie der 

 Tliêtafunctionen mil zwei Arijumenten, Crellk's Journal, vol. 94, {)age 

 75), which has the property that an arbitrary congruent transformation 

 of S^ can be replaced by hvo successive elemenlaiy (i-ansformations. 

 The name "Elementary rotation" (Elementardrchung) of Mr. Jahnkk 

 seems to me less fortunate, because it is a symmetric transformation, 

 not a rotation. The real meaning of the "Elementary transformation" 

 will be made clear furtheron. For Ihe present we remind the readers 

 of its determinaul lype (see JahresberichI, I. c., page 180). 



55 

 Proceedings Royal Acad. Amsterdam. Vol. VI. 



