370 DR. PLUCKER ON IIJNDAMENTAL VIEWS EEGABDING MECHA>rCS. 



the coordinates of Avhich are ^, u', iJ and t, u, v, one of the two planes (if, u', v') being 

 acted upon. The right line, along which both planes meet, is the axis of rotation. The 

 plane acted upon (^, u', v') may in a double way turn round the axis of rotation in order 

 to coincide with the second plane (t, ic, v) ; but there is no more ambiguity in admit- 

 ting that during the rotation the rotating-plane does not pass through the origin, and 

 consequently its coordinates do not become infinite. (In an analogous way we deter- 

 mine the distance of two points.) 

 Let us regard the six quantities, 



, , .. t—ff u—u', v—v, uil—u'v, vif—t^f, tu'—tfu, .... (2) 



as the six coordinates of the rotatory force, as they are the six coordinates of its axis of 

 rotation. As far as we do not regard the plane acted upon by the rotatory force, we 

 may replace them by the following six, 



X, ^, 3, ^, m, % (3) 



in admitting the equation of condition, 



S3£+3»3)+«»3 = (4) 



Finally, we may write the coordinates (2) in the following way, 



X, 8), 3, ^v-Su, 2t-3cv, Xu-m, (5) 



(f, u, v) being ani/ plane passing through the axis of rotation. 



2. The notation of the preceding number being rather unusual, it appears suitable 

 to introduce a few remarks before proceeding. 



In referring to the " Additional Notes " of the geometrical paper *, we get 



• - ' - X : Y : Z = cosX : cosfju : cos)', (6) 



L : M : N = cos a : cos/3 : cosy; (7) 



and in putting 



2e'+r+3'=*:ps (8) 



i'+m'+^'=m\ (9) 



there results 



91=1 • (10) 



Here the angles made by the axis of rotation with the three axes of coordinates OX, 

 OY, OZ are denoted by "k, ^,v; the angles made with the same axes of coordinates by 

 the right line perpendicular to the plane containing the origin and the axis of rotation 

 by 05, j3, y ; I denotes the distance of the axis of rotation from the origin ; finally, let 

 us call P the intensity of the rotatory force, K its moment, and the right line passing 

 through the origin and making, with the three axes of coordinates, the angles a, /3, y, 

 he axis of the moment. 



* PhilosopHcal Transactions, 1865, p. 776. 



