DR. PLUCKER ON FUNDAMENTAL VIEWS REGARDING MECHANICS. 379 



The axes of rotation belonging to a double congruency constitute a linear congruency 

 of right lines, represented by 



<I)-$'=0, 0-0"=O, 



and consequently meet two fixed lines. 



16. The axes of the congruent rotations oifour complexes are directed along the 

 generatrices of a hyperboloid. Hence we conclude that all axes of rotation are confined 

 within the tangent planes of the hyperboloid. Such a plane being given, its four con- 

 jugate points in the four complexes are within the same conjugate plane, intersecting 

 the given tangent plane along the axis of rotation which it contains. There is within 

 the given tangent plane a line of the other generation of the hyperboloid. When the 

 tangent plane revolves round this line, the corresponding axis of rotation, in revolving 

 simultaneously, in all its positions intersects the line in a point which describes it, while 

 the axis of rotation describes the hyperboloid. 



17. There are two rotations coincident in five complexes. 



18. The second kind of complexes of rotations is represented by the equation 



D3£+E8)+F3+A«+B3)Z+Cgi=l, 

 in regarding X, Y, Z, L, M, N, involving the condition 



as variable coordinates. All discussions regarding the new complexes are analogous to 

 former ones. 



19. In not admitting the last equation of condition, the complex of rotations of the 

 second kind is replaced by a complex of (rotatory) dynames. 



III. 



From the notions developed in Parts I. and II. we immediately obtain two general 

 theorems, constituting the base of statics. In a similar way, as D'Alembeet's principle 

 is derived from the " principe des vitesses virtuelles," both theorems may be transformed 

 into fundamental theorems of mechanics. 



Any forces acting upon a rigid body may be resolved into forces producing translation 

 and forces producing rotation. In the case of equilibrium, neither a translatory nor a 

 rotatory movement takes place, i. e. the resulting forces of both kinds become equal to 

 zero. 



In denoting the ordinary forces by 



(a/, y, 2*, X, y, z), 

 the rotatory forces by 



(i', u', v/, t, u, v), 

 the equations of equilibrium are 



X(x-a^)=0, %-y)=0, X(z-z')=0, 



X{t-f) =0, X{u-u')=0, t{v-v')=0. 



