234 MKSSRS. R THRELFALL AND J. A. POLLOCK 



a lever having mass, but of linear dimensions, fixed at the middle point of the thread. 

 Let the thread be considered uniform in diameter, and homogeneous. Let the centre 

 of gravity of the lever be at D, and at first consider the lever as hanging vertically, 

 so that D is vertically below the thread. Under these circumstances, let there be no 

 twist in either half of the thread. 



Let 6 be the angle through which the circle end of the thread, I, is rotated from 

 its initial position, in other words, let 6 be the twist in the circle end of the thread 

 when the lever is kept vertical in its initial position. 



Let <f> be the corresponding quantity referring to the spring end of the thread. 



Let / be the angle which the lever makes at any moment with the vertical plane 

 drawn downwards through the thread. 



Let I be the distance of the centre of gravity of the lever from the thread. 



Let m be the mass of the lever. 



Let g be the value of the earth's acceleration at the point considered. 



Let T be the moment of the forces exerted on the lever by either half of the thread 

 when twisted through unit angle. 



The equation of equilibrium, neglecting for the present the effect of variations of 

 temperature, is 



mgl sin \}i = T (0 \ji) + T (<f> \jj) (1). 



If the system is in equilibrium and we increase 6, the lever will take up a new 

 equilibrium position, provided that is finite. If -~ is infinite any increase of 

 will make the lever upset. Now, from Equation (l) above 



so - - will be infinite when 



(1(7 



dd mgl cos ty + 2r 

 mgl cos $ -f 2r = 0, 



i.e., when 



2r 



cos tl = -- . 

 mgl 



Also from the original equation 



d-Jr 

 so -J will be infinite when 



sn 



In the case of the instrument as constructed, we know that (0 \jj) and (< t//) 

 are both approximately equal to GTT, so that the upsetting position of the lever is 



approximately given by cot i/ = , or the lever is above the horizontal plane 



, 



