1880.] 



Torsion- Gravimeters of Broun and Babinet. 



143 



Here A is an instrumental constant, which defines the relation of the 

 parts 9, 0, of the whole angle rj. As rj increases, 9 increase in the 



proportion JL cos 9. Suppose a point reached where ceases to 



increase: then — cos 9=0, and 0=90°. At this point, any increase 



of rj is wholly absorbed by 9 ; which is easily understood, because at 

 this point the bifilar opposition to torsion is at its maximum. Next, 

 suppose ?/ to be further increased, until the rate of decrease of equals 

 the rate of increase of 9 • in other words, to the utmost consist- 

 ent with the relation between these components. At this point, 



^= A. cos O == — 1, or cos#q= — A. At this point, too, 9 can vary 



reciprocally without affecting ?/. It is a position which can only 

 incorrectly be described as one of unstable equilibrium ; for though 

 the weight would not return if moved forward, but on the contrarj^, 

 would continue to move forward, under the pressure of torsion; yet, 

 if moved backward, it would seek to return. It is therefore stable on 

 one side, and unstable on the other. 



It is easy to test this practically. I have done so, and recognise, in 

 the peculiar conditions, such as are well suited to afford an exact 

 determination ; as I will explain presently. But there are one or two 

 theoretical points to be first noticed. 



Since cos 9 = — A, it follows that 00= — tan 9 ■ that A must be made 

 not greater than unity ; and that 9 must lie between 90° and 180°. 



The condition that P must be >— — T is an important one. When- 



ever this is the case, there must be a value of 9, at which the peculiar 

 equilibrium will occur. The condition may be fulfilled in a way to 

 make the position unsuitable for exact observation, but it nevertheless 

 exists ; whereas, if P is too small, there is no such position. 

 Transposing, we see that — 



1 T 



r 2 must be > — , 



7T P' 



from which it follows that whatever be the strength of the wires, or 

 quantity of the weight, it is always possible, by modifying the 

 distances between the former, to secure the necessary condition. 



From this point of view, too, it appears that the length of the 

 wires is immaterial. The opposing forces are equally affected by a 

 change of length. The magnitude of R therefrom has nothing to do 

 with the equilibrium. 



Let us now work out a case. I find that a piece of pianoforte wire, 

 0'03 inch diameter, will carry 100 lbs. ; will bear, without apparent 



M A 



