378 DB. PLtJCKEE ON FUNDAMENTAL VIEWS REGARDING MECHANICS. 



exclusively rotations of this description, the six coordinates of which are likewise infinite, 

 the equation of the complex becomes 



H=0. 



Being now homogeneous, it represents a linear complex of axes or right lines, identical 

 with the complex represented in Part I. by the equations 



Q = 0, or 4)=0. 



It would be beyond the limits of this paper to develope here the theory of linear com- 

 plexes of rotations. Let me observe only that, in taking for OZ the axis of the complex 

 H, which may be regarded likewise as the axis of the complex of rotations 0, the 

 general equation of the complex assumes the following form, 



v—v'-\-x(tu'—tfu)=xx', 



in denoting by x and x' two constants, and may in retaining the former notation be 



written thus, 



„/ cosy\ , 



r( COSP+* .—j^]=xx. 



There are amongst the rotations of the complex such transformed into translations. 

 They will be determined in putting S = oo , whence 



P cos )»=:««'. 



14. The congruent rotations of any two complexes, 



0=0, 0'=O, 



constitute a congruence of rotations. Any plane being given, there is in each complex 

 a point conjugate to the plane ; the line joining both points may be called conjugate 

 in the congruence to the given plane. Each plane passing through the conjugate line 

 intersects the given plane along an axis of rotation. Therefore all axes within the plane 

 meet in the same point, where it is intersected by its conjugate line. Among the axes 

 there is one confined in the plane passing through the origin ; in the corresponding 

 rotation P becomes infinite. Again, there is one rotation transformed into a displace- 

 ment parallel to the given plane. 



In accordance ^vith these results, the equation 



0-0'=O, 

 derived from the preceding ones by eliminating their constant term, represents a liriear 

 complex of axes. 



15. The congruent rotations of three complexes, 



0=0, 0'=O, 0" = O, 

 constitute a double congruency of rotations. Any plane traversing space being given, 

 there is another plane passing through the three points conjugate in the three com- 

 plexes to the given one. This plane may be called conjugate in tlie double cmigruency 

 to the given plane. There is within the given plane one single axis of rotation coinciding 

 with the intersection of both planes, that given, and its conjugate one. 



