368 DE. PLUCKEK ON TUXDAMEXTAL VIEWS EEOAEDING MECHANICS. 



not enter into any detail, because the results thus obtained would be involved in the 

 following developments. 



12. Indeed, in so transforming the arbitrary system of rectangular coordinates — as 

 we did in the case of complexes E — that the new axis OZ coincides with the axis of the 

 complex of lines 4>, the equation (24) is replaced by the equation 



■^^k(Z-l') = 0, (28) 



k and /{f retaining their signification of No. 4, and may be written thus, 



F{hcosv+kcosy)=Uv', (29) 



in denoting by y and c the angles which the directions of the forces and of the axes of 

 their moments make with OZ, and by I the distances of the lines along which the forces 

 act from the origin. Hence we conclude that the intensities of forces of the complex 

 are the same if J cos. + A" cosy = const (30) 



That is especially the case if the line along which a force acts be displaced parallel to 

 OZ or turned round it. Hence 



A force of a complex "^ lohkh, while retaining its intensity, is displaced imrallel to the 

 axis of the complex or turns round it, in all its new positions continues to belong to the 

 cwnplex. 



13. The lines along which congruent forces of any two complexes ■*" act constitute 

 a linear complex of lines. The congruent forces of three complexes "V are directed 

 along lines of a congrucncy, and consequently meet two fixed lines, there is one force 

 passing through each point of space, and one confined within each plane traversing it. 

 The congruent forces of four complexes "^ are directed along the generatrices of a 

 hyperboloid, their intensity only varying from one generatrix to another. Finally, five 

 complexes "i' meet along two forces (either real or imaginary). 



14. A dyname, determined by its six linear coordinates, 



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



represents the eff"ect produced by two forces not intersecting each other, the pomts 



acted upon not being regarded. The six sums of the corresponding coordinates of both 



forces are the six coordinates of the dyname. Keciprocally, a dpiame, the coordinates 



of which are given, may be resolved into equivalent pairs of forces ; but a dyname 



depending upon six, a pair of forces upon ten constants, four of these ten constants may 



be chosen arbitrarily. Let , ^ ,, „ 



•' x,y,z, L,jr,>, 



x', y', z', l', m', k', 



be the coordinates of such a pair of forces. The following relations, 



x+x'=X, y+y'=Y, z+z'=Z, 

 l+l'=L, m+m'=M, N+if'=N, 



