ACOUSTICS AND GRAVITATION. 135 



between 0.39 and 0.45 (if the exceptional results are excluded), depending on 

 changes of temperature outside of the laboratory. The two graphs, morning 

 and afternoon results, as a whole are similar. 



After removing the needle, the torsion coefficient of the quartz fiber was 

 found by the vibration method, two small brass cylinders of moments of inertia 

 AT=o.o4g8 and 0.0486 being used in succession, the periods being 7=5.51 and 

 ^=5-39 seconds. Unfortunately, both these data are rough, for it was found 

 difficult to make the small cylinders rotate axially quite at will. Accepting the 

 mean temporarily, the value of the constant 7 may be then computed as 



y 



from the double deflection Ay, if R = 4.2 cm. is the distance between the center 

 of the masses M=Q49 grams and m= 0.581 gram, L = 26i cm. the scale-dis- 

 tance, and /=n cm. the semi-length of the needle. The mean value thus 

 obtained was 



7=io~ 8 Xi8.2 Ay 



so that for y = io~ 8 ,X6.66 the deflection should have been 0.37 cm. Even the 

 vacuum values are thus excessive, except on the dark days, September 30 and 

 October i ; but it is probable that with allowance for the stem attraction, the 

 data for Ay found will not be far from the correct values. At all events, it is 

 clear that observations made with the needle in vacua show deflections which 

 are an enormously closer approximation to the truth than the plenum values, 

 and that further pursuit along these lines will probably lead to trustworthy 

 data. The mean a. m. and p. m. displacements were Ay =0.44 cm. and Ay = 

 0.43 cm. This is an excess of less than 16 per cent, most of which is referable 

 to the attraction of the stem of the needle, as appears in the next paragraph. 



102. Torque exerted on the stem. If the center of the mass M is at right 

 angles to the needle of semi-length /, and at a distance R from it; if * be 

 measured from the end of the needle to its center (* = /), and if dx be at a 

 distance r from the center of M, it is easily seen that the torque will be 



l MpdxR 





,, . 



*> 



where p is the mass of stem per unit of length. The integral^of this expression 



is 



or 



t=(yMp/R) 



where the coefficient yMp/R is the total attraction in the direction R for a 

 long needle. 

 IfT=(vMm/R*)l, then, 



