( 727 ) 

 to OPr in Sr in such a way that 



rOPr,OW\ 



V Oa, OSr ) 

 is equiangular to the right, then to the equiangular double rotation 

 to the right of S^, indicated by OPj. coi-responds the rotation of S,- 

 about OPr equal to the length of OPr in the direction of OQr -^ OSr. 

 Let OP'r be another indicating vector and let us determine in an 

 analogous way OQr and OS'r, then the orthogonal systems of vectors 



OWPrQrSr and 



OWF.QrS'r 



are congruent, can thus be made to coincide by a rotation of S^, 

 with OW, thus Sr too, remaining in its place, so that the indicating 

 vector OPr by a motion of Sr in itself can be made to coincide 

 with the indicating vector OP'r in such a way, that at the same 

 time the directions of rotation of Sr belonging to it coincide in the 

 normal planes. But then an indicating vector in Sr represents that 

 velocity of rotation entirely in the way usual in space of three 

 dimensions, as also by its length it indicates the size of the velocity 

 of rotation of Sr belonging to it; if namely we endeavour to regard 

 the definition of that usual mode of representation entirely apart 

 from the notion ''motion with or against the hands" which is lacking 

 in /S4 ; and say simply after having taken in that space a system 

 of coordinates OXYZ: the vector of rotation will be erected to 

 that side of the plane of rotation, that for the plane of rotation 

 being by motion inside the space made to coincide with the plane 

 of XY in such a way, that the direction of rotation runs from OX 

 to OY, the vector of rotation is directed along the positive axis of -.^. 

 Yov us that system of coordinates in Sr'- OXr, OYr, OZr has 

 been chosen in such a way that with OW \i forms a system of 

 coordinates in S^, for wdiich 



OYr, 0Z,\ /OZr, OXr\ /OXr, OYr' 

 PXr, OW/ yOYr, OW/ \OZr, OW ^ 



are equiangular to the right. 



A vector along OXr then indicates a rotation of Sr in the sense 

 of OYr -^ OZr. 



Entirely analogous reasonings hold good for S[. It being however 

 more profitable to be able to say also for Si: a vector along OXi 

 represents a velocity of rotation of S[ in the sense of Yi -» OZi, 

 we must modify the preceding either by choosing the system 

 OXi Yi Zi W in such a way that 



