516 THE THEORY OF SCREWS. 



He then considers the quaternion ? ^73 already mentioned (p. 512), and 



i/3 



&quot;S? /? 



introduces the new quaternion Q = -- = c + y. The scalar c (the pitch) is inde- 



Z/3 



pendent of the assumed origin, and the vector y is the vector to a definite point C 

 on the central axis. This point does not vary with the position of the assumed 

 origin, and is called the &quot;Centre of the System of Forces.&quot; When the forces are 

 all parallel G coincides with the centre of the parallel forces. In general 



or the tensor of the total moment is constant for all points situated on a sphere 

 whose centre is C, and becomes a minimum when coincides with C. 



In Art. 396 Hamilton says &quot;the passage of a right line from any one given 

 position in space to any other may be conceived to be accomplished by a sort of screw 

 motion&quot; and on these kinematical lines he worked out his theory of the &quot;Surface of 

 Emanants,&quot; generated by a line moving according to some given law and constantly 

 intersecting a given curve in space. 



PLUCKER (J.) Fundamental views regarding mechanics. Phil. Trans. (1866); Vol. 

 clvi., pp. 361380. 



The object of this paper is to &quot;connect, in mechanics, translatory and rotatory 

 movements with each other by a principle in geometry analogous to that of re 

 ciprocity.&quot; One of the principal theorems is thus enunciated: &quot;Any number of 

 rotatory forces acting simultaneously, the co-ordinates of the resulting rotatory 

 force, if there is such a force, if there is not, the co-ordinates of the resulting 

 rotatory dyname, are obtained by adding the co-ordinates of the given rotatory 

 forces. In the case of equilibrium the six sums obtained are equal to zero.&quot; 



SPOTTISWOODE (W.) Note sur Vequilibre des forces dans Fespace. Comptes 

 Rendus; Vol. Ixvi., pp. 97-103 (January, 1868). 



If P ... P re _! be n forces in equilibrium, and if (0, 1) denote the moment of 

 P , P 15 then the author proves* that 



^(0, l) + P f (0, 2)+...=0, 



P (2, 0) + P 1 (2, 1) + 



As we have thus n equations to determine only the relative values of n quantities, 

 the redundancy is taken advantage of to prove that 



p 2 p 2 



[ofo] = [17T] = &c &quot; 



where [0, 0], [1, 1], &c., are the coefficients of (0, 0), (1, 1), &c., in the determinant 



(0, 0), (0, 1) ... 

 (1, 0), (1, 1) ... 



* We may remark that since the moment of two lines is the virtual coefficient of two screws 

 of zero pitch, these equations are given at once by virtual velocities, if we rotate the body round 

 each of the forces in succession. 



