362 DE. PLUCKEK ON FUNDAMENTAL VIEWS EEGARDING MECHANICS. 



But a force depending upon five constants only, there exists between these new coor- 

 dinates an equation of condition, namely, 



LX+MY+NZ=0, (3j 



which indicates that the axis of the resulting moment R (the moment of the force mth 

 regard to the origin) is perpendicular to the direction of the force. In replacing the 

 coordinates (2) by the equivalent primitive ones (1), the last equation becomes an iden- 

 tical one between the six point-coordinates o^, y , z', a:, y, z, and therefore is involved in 

 the form given to the coordinates of the force. 

 The three last coordinates, 



yz'—i/z, za^—z'x, x^—o^y, 



remain unchanged in replacing x, y, z by [x—a^], {y—y')-, (2—2'). Consequently we may 



substitute for them 



Yz'-Zy, Z^'-Xz', xy-Ya/. 



Thus, in omitting the accents of a/, y, z', 



X, Y, Z, Yz-Z//, Z^-Xz, Xy-Y^ (4) 



become the coordinates of the force. Now x, y, z denote the coordinates of any point 

 of the line along which the force acts, its intensity and direction being given by X, Y, Z. 

 The form of the new coordinates (4) involves the equation of condition (3). 



2, If any number of given forces, represented by the symbols (a/, ij , z', x, y, z) or 

 (X, Y, Z, L, M, N), act upon or pass through given points, according to the fundamental 

 laws of statics, the resulting effect is obtained by adding the corresponding six coordinates 

 of the forces 



x—x', y—y', z—z', yz'—'i/z, zx'—z'x, xy'—sdy. 



If the six sums thus obtained, 



t{x-^^\ S(2/-y), %{z-^\ 2(yz'-yz), ^x-^x), t{xi/-a^y), . (5) 

 or 



SX, 2Y, XL, SL, SM, ^N, (6) 



satisfy the condition 



§L.SX-fSM.SY+2N.:^;z=0, 



and therefore assume the form of the expressions (1), they are the six coordinates 

 of a resulting force which replaces the given ones. In the general case I propose to 

 call the cause producing the resulting effect dyname. The six sums (5) or (6), not 

 satisfying the last equation of condition, may be regarded as the six coordinates of the 

 dyname ; the first three indicating the intensity and the direction of a force P, the last 

 three the intensity of a moment and the direction of its axis. 



In the case of a force (P) depending upon five constants, the moment and the direc- 

 tion of its axis are determined by means of these constants, i. e., by means of 



X, Y, Z, L, M, N, . . (2) 



in admitting 



LX-i-MY+NZ=:0. 



