380 DB. PLUCKER ON FUNDAMENTAL VIEWS EEGAHDING MECHANICS. 



In putting, n being the number of forces, 



Sa/=«r, '^i/=m\ lz!=n^, 



S^=w^, 2M'=wg', 2t/=«o^, 



(I', ;;', ^') and (|, t;, ^) are the centres of gravity of the two systems of points (a/, i/, :!) 

 and [x, y, z) ; likewise (^', §', (/) and (Sy, g, ff) the central planes (II. No, 9) of the two 

 systems of planes (^', v! , v') and {t, u, v). Accordingly the equations of equilibrium 

 become 



%z=%\ ^=/, ff=ff'. 



We commonly represent ordinary forces by means of right lines, analytically by means 

 of the coordinates of their extremities, i. e., by the coordinates of the points acted upon 

 (of, y, z') and the coordinates of second points (a; y, z). In an analogous way rotatory 

 forces are represented by axes and couples of planes passing through them ; analytically 

 by the coordinates of planes acted upon {U, u', i/), and the coordinates of second planes 

 (t, u, v). Accordingly in the case of equilibrium — 



I. The centre of gravity of the points acted upon hy the forces coincides with the centre 

 of gravity of the second extremities of the right lines by which the forces are represented. 



II. The central plane of the planes acted upon by the rotatory forces coincides with 

 the central plane of the second planes, by which these forces are determined. 



If we introduce the notion of masses both theorems hold good, only the definition of 

 both kinds of forces and therefore their unity is changed. The points acted upon 

 become centres of gravity, corresponding to masses ; the planes acted upon central planes, 

 corresponding to moments of inertia. 



If equilibrium does not exist, there is in the general case one resulting ordinary force, 

 determined by the two centres of gravity, and one resulting rotatory force, determined 

 by the two central planes. The intensities of the two forces are 



These forces decomposed into three are known, and therefore the direction of the 

 axes, both of translation and rotation. We get easily the six differential equations of 

 the movement produced. 



I shall think it suitable further to develope the principles here merely indicated. A 

 Treatise on Mechanics, reconstructed on them, will assume quite a new aspect. 



