JoLY — Asiatics and Quaternion Functions. 369 



wrenches of given pitch are in this canonical form concentric, and 

 their equations are 



SK{(f> - c) K = e - c. 



Some account is also given of the conditions that a body having 

 various degrees of freedom may be in equilibrium under the action of 

 the forces, and of the manner in which the forces may be turned 

 without disturbing the equilibrium of the body. 



When dealing with the effect of the force systems when a point of 

 the body is fixed, it may be more convenient to use Hamilton's second 

 equation 



2a/? = C+ IX. 



This, when the forces are turned, furnishes the relation 



Fi={C+,.)S. (D) 



And, when the body is turned, 



Fq = q{C+f.). (E) 



In these, Fq = 2a§'/5, and /jl is the couple, and - C the virial of 

 the rotated force system for the fixed point as base-point. 



Comparing these with (A) and (B), we see that /jl and C are related 

 to i^in precisely the same manner as y and c are related to/. 



For example, from (D), 



fM = VFqq-\ 



so if lines are drawn through the origin to represent the resultant 

 couples, their extremities fill a certain region ; the points corresponding 

 to a given direction about which the rotation is performed lie on an 

 ellipse. Also, the points corresponding to a given value of the virial 

 lie on a cyclide ; and, with new interpretations, many of the results 

 stated concerning the former equations (A) and (B) apply equally to 

 these new equations (D) and (E). 



