94 NOTE ON THE COMPOSITION 



"When this condition is satisfied, (provided 2(X), 2(Y), S(Z), do 

 not all vanish,) the equations (I) are equivalent to only two inde- 

 pendent equations, and represent a straight line, every point of 

 which is an origin such as required. 



In the particular case where the system consists of Forces in 

 parallel directions, taking F as the type of one of these at the 

 point (x, y, z), and I, m, n for the direction-cosines of their common 

 direction, we have 



2(X) = 1%(F); 2(T) = m%{F)\ S(Z) = »5(J); 

 L = n 5(%) — m 2(Fz) 

 M = I 2(Fz) — n %(Fx) 

 N = w2(Fa?) — I S(Fy) 



The condition (2) is in this case satisfied, and (provided 2(jP) do 

 not vanish) the equations (1) assume the form 



£ m n 



Hence the line of action of the siugle Resultant passes through 



■ 4. i v i. 2(Jfc) , %{Fy) , %{Fz) v: 



the point whose co-ordmates are Tr ,.- rT . _ , J;/ ^, -n/ ; these 



* %{F) %{F) %{F) 



are independent of I, m, n, and this point therefore remains the 

 same so long as the forces and their points of application are unal- 

 tered, whatever be their direction ; for this reason, it is called the 

 Centre of Parallel Forces. 



In like manner, the motion at any instant of a free rigid body 

 can be reduced to a single rotation about an axis passing through 

 some assigned point as origin, and to a single motion of translations 

 proper to this origin and common to all the points of the body; the 

 former of these remaining invariable both in magnitude and direc- 

 tion, whatever origin be assumed, while the latter varies in both 

 respects for different origins, remaining constant, however, for 

 origins situated along the axis of Rotation. 



Adopting the usual notation by taking u> x , w , io s , for the compo- 

 nents of the rotation round three rectangular axes, and u, v, w for 

 the components of the velocity of translation along the same axes, 

 we have for the velocities u' , v' , w,' along these axes, of a point 



v' '~ °^y' 



u' 



= 



u 



+ 



1/ 



== 



V 



+ 



id 



= 



w 



+ 



