DK. PLiJCKER ON FUNDAMENTAL VIEWS REGARDING MECHANICS. 367 



indicate the direction of any of these forces, we obtain 



X=«Z, Y=JZ, 

 whence 



1 



Z= 



and the intensity of the force 



Aa + Bb+C' 





If the system of coordinates is displaced parallel to itself, any point (Xo, y„, Sq) becoming 

 the new origin, X, Y, Z remain unaltered, while x, y, z are replaced by {x-\-Xo), (^+^0)? 

 (z+Zj). Accordingly the equation (25) is transformed into the following one : 



AX+BY+CZ + D(Y(z+z„)-Z(y+y„))+E(Z(x-f^o)-X(z+^o))+F(X(y+y.)-Y(^+a:.)) = L 



In putting .r, y, z=0, the following relation 



(A-Ezo+Fj/.)X+(B+D.^„-F^„)Y+(C-D^.+Rro)Z=l . . . (27) 



is obtained between the coordinates of forces passing through the new origin. Let, in 

 the primitive system of coordinates, 



a;— j'„=a(z— z„), 



y—yo=Kz—z,) 



indicate the du-ection of any such force, its intensity is 



~ (A - Ezo + F^o) « + (B + D^o- Fa^o) 6 + (C - D^o + Ea^o) 



._ 1 



~ (A - E^o + Fyo) cos a + (B + D^o - Fa^o) cos /3 + (C - Dj^o + Ex^) cos 7 



_ 1 



A cos « + B cos /3 + C cos 7 + D(^oCos |3 — ?/qCos y) + E(xq cos y—s^ cos «) + F{yQCOs a— .TqCos j3)' 



There is one force passing simultaneously through both origins, determined by the 



relations 



^0 •■ yo : 2!o = X : Y : Z, 



by which the last expression for P is reduced to the former (26). The force acting 

 along the same right line is the same. All other forces of the complex T" passing 

 through the primitive origin, when displaced parallel to themselves, so as to meet 

 the new origin, generally change their intensity. This intensity is not changed if, the 

 direction of the force remaining the same, 



Ez„=Fy„, 



T), —Ft 



D^„=E.ro, 

 i. e. if the new origin describes what we may call a diameter of the complex. We do 



