210 Mr. T. Gray on Instruments for Measuring 



attached to it the index i (which is hinged by means of a strip 

 of tough paper at h, and rests through a fine pin on the glass 

 plate g), and is partly filled with mercury. 



The reasons for this mode of suspension are :— First, the 

 arm A, by allowing the spring to be held stretched by means 

 of a lever and weight instead of by a weight directly applied, 

 increases the period of free vertical oscillation of the spring 

 and weight. For let (j> = angle turned through by the arm A 

 from its normal position at time t, e = consequent elongation 

 of the spring, E = total normal elongation of the spring, 

 I = length of long arm of lever A, V= length of short arm 

 of A, g = force of gravity in unit mass, T = period of ver- 

 tical vibration, M the mass of the lead weight E,— then, 

 supposing (/> small, the lever A very light, and the mass of R 

 collected at its centre of inertia, and neglecting the influence 

 of the trough &c, we have for the equation of motion, 



and therefore 



*± + 2.1 4,-0 



From this we see that the period increases as the square root 



of y ; and hence an advantage in length of period is gained 



by attaching the string as shown. A disadvantage, of course, 

 is that a smaller mass (in the ratio of V to I) is required to 

 produce a given normal elongation of the spring than if the 

 weight were applied directly. 



Secondly, the mercury in the trough t acts as a compensator 

 to prevent the ring R moving when the top of the spring is 

 moved through a distance short compared with the distance 

 between the pivots on the trough t. The action of this part 

 is as follows: — When the plane carrying the spring S is raised 

 and lowered, the point a rises and falls, but in consequence of 

 the inertia and slow period of R the point C remains behind. 

 In consequence of this the end of the trough t falls and rises 

 relatively to a ; and the mercury, running backwards and for- 

 wards, puts more or less force on the point C, and hence tends 

 to keep this point stationary. 



If the length of the trough be x, the distance between the 

 pivots y, and the width w, then for a rise of a through a dis- 

 stance h we have the centre of gravity of a prism of liquid of 



depth h moved from the centre of the trough to a point o from 



o 



