152] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 141 



These will evoke wrenches on E and F of the intensities 



HF HE 



EF EF 



respectively. But this pair of wrenches are to compound into a wrench of 

 intensity h on H , and consequently we have 



H F HF 



H E _ 



S EF 



HF H F , 

 whence ~ : , :: J : e. 



If we take another pair of points, K and K , we have 



KF H F K F 



^ 

 HE : KE :: H E K E 



whence (HKFE) = (H K F E ). 



Thus, the anharmonic ratio of any four points in one system is equal to that of 

 their correspondents, and the two systems are homographic. 



The homographic axis intersects the circle in two points, which are the 

 principal screws of the potential, i.e. a twist about either evokes a wrench 

 on the same screw. Of course this homographic axis is distinct from that 

 in 139. But this homographic axis, like the former one, passes through the 

 pole of the axis of pitch because the principal screws of the potential are 

 reciprocal. 



152. Work done by a Twist. 



Suppose that the body, when in equilibrium under the system of forces, 

 receives a twist of small amplitude of about any screw a, a quantity of work 

 is expended, which we shall denote by 



Fv a -a 2 . 



In this, F is a constant, whose dimensions are a mass divided by the square 

 of a time, and v* is a linear magnitude specially appropriate to the screw a, 

 and depending also upon the system of forces ( 102). We may compare 

 and contrast the three quantities,^, u a , v a : each is a linear magnitude 

 specially correlated to the screw a. The first and simplest, p a , is the pitch 

 of the screw, and depends on the geometrical nature of the constraints ; u a 

 involves also the mass of the body, and the distribution of the mass relatively 

 to a ; v a , still more complicated, depends also on the system of forces. 



