DE. PLUCKIEE ON FUNDAMENTAL VIEWS REGAEDING MECHANICS. 377 



of coordinates. The three movements thus obtained give a resulting movement of the 

 same kind. 



If only the first three of the six equations are satisfied, ir, x, g becoming equal to zero, 

 the resulting axis of rotation passes through the origin. 



13. After this digression, by which a full analogy between ordinary forces and forces 

 producing rotation is stated, we may proceed by giving most succinct indications only. 



With regard to rotations and forces producing them, we have to distinguish two dif- 

 ferent kinds of complexes corresponding to their difierent systems of coordinates. We 

 shall first, in making use of the coordinates 



t—if, u—u', v—iJ, uif—u'v, vt'—'dt, tu'—Hu, 



consider a complex of rotations, the coordinates of which satisfy the equation 



which, for brevity, may be written thus, 



0=H-1 = O. 



In regarding #', w', «' as constant, any fixed plane traversing space is the plane acted 

 upon by the rotatory forces, and therefore containing the axes of rotation. The coor- 

 dinates of the second planes {t, u, v), by means of which the corresponding rotations of 

 the complex are determined, remaining variable, the same equation representing the 

 complex now represents a point, where all second planes meet. The equation of this 

 point may be written thus, 



(D+Cm'-Bj/)« 



(E-a' +A«;> 



(F+B^ -Av!)v 



whence the following coordinates of the point are obtained, 



D + Cm'-B»' 



x=. 



D^'+Ew' + FtZ-l' 



_ E-C i!' + A»;' 

 ^— D/' + Em'+Fi/-!' 



r+B/'-At/_ 



^— D^ + Em' + F«'-1' 



We shall call this point the point conjugate to the plane [il, u', v'). 



Any plane traversing space may he regarded as acted upon by the forces of the complex, 

 each right line it confines, as an axis. The rotation corresponding to each axis is deter- 

 mined by a second plane, traced through the conjugate point and the axis. 



The intensity P of each rotatory force is thus immediately given. P becomes infinite 

 for all rotations the second planes of which pass through the origin. In considering 



MDCCCLXVI. 3 F 



