On Gravimeters. 



521 



To understand thoroughly the conditions of equilibrium' we must 

 now study in detail the counteracting forces. We have seen that the 

 weight supported by the double wire is the sum of the two weights 

 described as the major and the minor weights. One-half of this sum 

 is, of course, borne by each of the wires, which are further twisted in- 

 dividually through a quarter of a circle. Let R be the length of these 

 wires, and 2r their distance apart. Consider one only : the upper end 

 being- fixed, the lower will describe a curve which will be the inter- 

 section of a sphere by a vertical cylinder, the ordinate from which to 

 a plane drawn horizontally through the lowest point may be shown 

 to be equal to 



r . versin 6 1 1 + i^> • versin 6 — &c. | 



As r : R is a small fraction, less than y^, it will be a question whether 

 the second and farther terms may not be neglected ; for the present 

 we may be content with the first, as an approximation only is wanted 

 here. This ordinate is the height through which the weights rise as 

 they are turned round. Let this be called h. A little consideration 



of the meaning of -- — =— , a constant, shows that in this case the 



r versin R 



restriction to the first term makes the path of the lower end of the 

 wire a circle lying in an inclined plane,* and the force tending to 

 cause rotation (irrespective of the torsion of the parallel wires) is that 

 of a body having the joint weight of the two, whose path of descent 

 is to be along this circle. The gradient along this path (which is the 



tangent of the inclination) is — — — sin 0, which is zero at 0° and 

 & r.ddB, 



180°, and a maximum at 90°. Let A + B = P be the weights, which 

 are augmented for adjustment by a small weight, p. Then the force, 

 depending on gravity, which tends to turn the system, resolved hori- 

 zontally, is (P+jp) tan Z inclination = (P-fp)-^-sin 9. 



R 



Neglecting for the present the torsion of the parallel wires, we see 

 that this force is resisted by the torsion of the single wire, as to which 

 we must have regard to two propositions : — 



(1.) Torsion is independent of tension; 



(2.) Torsion varies directly as the angle of twist. 



We may, therefore, express the torsion by ^0 (at the distance unity), 

 remarking that is mechanically increased until it balances the force 

 tending to turn the system as above described. 



Now, since (V+p) — sin# attains a maximum when 0=90°, if at 

 R 



* This demands an elliptical cylinder. 

 VOL. XXXII. 2 P 



