DE. PLUCKER ON FUNDAMENTAL VIEWS REGARDING MECHANICS. 375 



may be decomposed into three, 



{t'v!v\ t'u'v), (tfu'i^, t'uv'), (fu'v', tic'v')*, 



the six coordinates of which are 



v-r/ -u'(v-v') t(v-v') 



u-u' v'(u-u') -t'{u-u') 



t-a -v\t--i) ii(t-l!). 



In adding these coordinates, we get 



t—i, M— w', V — «', Mj;'— M'r, vH—v't, tu'—t'u, 



i, e. the coordinates of the recomposed given rotatory force. 



The three axes of the decomposed rotatory forces are the intersections of the plane 

 acted upon by (^, u', t/), with the three planes of coordinates XY, XZ, YZ, constituting 

 a triangle, the angles of which fall into the three axes of coordinates. 



The given rotatory force is thus decomposed into three equivalent ones, the intensity 



of which is 



$cosv ={v-v')=2i, 



5P cos /A =(%—%')= 3), 



5Pcosx=(^-^)=3. 

 In putting 



r, q, p denote the distances within the planes XY, XZ, YZ of the axes of the decom- 

 posed forces from the origin, and 



3, f, I 



r q p 



represent the three corresponding moments. These moments do not change if, 

 within the planes of coordinates, the axes of rotation revolve round the origin, and 



especially become parallel to an axis of coordinates; ^ for instance, if the corresponding 



axis become parallel to OY or OZ, is equivalent to one single coordinate ^^^, replacing 

 both former ones — u'(v—v') and t'(v—v'). 



12. Any number of rotatory forces being given, by decomposing each into three, the 

 axes of which are confined within the three planes of coordinates, and by recomposing 



* The decomposition and rocomposition of rotatory forces acting upon a given plane, as well as of ordinary 

 forces acting upon a given point, is immediately derived from the principle of the coexistence of infinitesimal 

 movements, whicli may be replaced by the causes producing them, i. e. by forces. 



