( 726 ) 



The plane FOW giving OP as representant of its equiangular 

 system to the right before its double rotation, gives after that rotation 

 (transferred to P' OW) as representant OP" making an angle 2y 

 with OP; so an arbitrary vector OP in Sr-i-OC (the invariable 

 vector) rotates about OC over an angle 2(p, with which the theorem 

 is proved. 



We can now say : For an S^ moving as a solid about a fixed 

 point the position is at every moment determined by its "position 

 to the right" (the position of the S,- moving as a solid about a 

 fixed point) and its "position to the left" (the position of Si). For, 

 if of two positions the pairs of planes through coincide, then this 

 is the case too for all planes, thus for all rays too. 



N.B. We can remark by the way, that in this way we have 

 proved quite synthetically that two positions of S^ have a common 

 pair of planes, namely that pair, which has as representants the 

 axis of rotation of the two positions to the right and that of the two 

 positions to the left ; so, taking into consideration that also the 

 common fixed point is always there (having as projections on the 

 positions of planes remained invariable the centres of rotation of 

 the projections of S^ on it), that the double rotation is the most 

 general displacement of S^. However, until now we have occupied 

 ourselves only and wish to keep doing so with the motions of S^, 

 where always the same point is in rest. 



For a continuous motion of S^ the position and the condition of 

 motion are at every momejit determined by Sr and Si ; so the motion 

 of S^ is quite determined by the motions of Sr and Si ; and at every 

 moment the resulting displacement of S^ is quite determined by 

 that of Sf and of Si, independent of the order of succession or 

 combining of the two latter ; they have no influence upon each 

 other. We can regard a motion of S^ as a sum of two entirely 

 heterogeneous things, i. e. which cannot influence each other in any 

 way or be transformed into each other. 



We can proceed another step by indicating not only by a line in 

 Sr a system of equiangular planes of rotation to the right, but 

 also by a line vector along it an equiangular velocity of rotation 

 to the right, that line vector being equal in size to the double 

 velocity of rotation of the double rotation and directed along the 

 vector moving with S^ in the direction of OW. Then equal and 

 opposite velocities of rotations to the right of S^ are indicated 

 by equal and opposite vectors in ^S,.. 



Let OPr be such an indicating vector and OQr and OSr two 

 mutually perpendicular \'ectors in the plane erected perpendicularly 



