NEWTONIAN CONSTANT OF GRAVITATION. 61 
reason of this is made clear if a correction, @ in terms of S, is included in the expres- 
sions for U and §._ Since the fibre becomes stiffer when the balls are taken off, it is 
simpler to consider this correction as a stiffening of the fibre when it is unloaded 
instead of the reverse when it is loaded. The expressions are now :— 

Ua Biv +45) 
op aso |S): 
‘ 4a? B 
S = Ta? — T2(D OS)” 
It now appears that in both, the denominator is affected to the same extent, which 
is very small since T,” is small compared with T,”?. On the other hand the numerator 
of S is not affected, while the correction applies to the whole numerator of U. 
The very great effect of this upon the absolute value of S is well shown in Experi- 
ment 9, where the additional weight was 7°975 instead of 5°314 grms. In con- 
sequence of the extra amount of stretching S$ fell from ‘001196 to 001147, or nearly 
5 per cent. The actual elongation of the torsion fibre in the two cases was ‘0394 
and ‘0677 inch. The whole length of the fibre was 17 inches, so the amount of 
stretch was ‘232 and ‘398 per cent. Even if the volume of the fibre remained 
constant the diminution of torsional rigidity could not be accounted for with a 
material of constant rigidity. This point is perhaps worth considering in connection 
with Potsson’s ratio and the theory of elasticity, more especially in consequence of 
the great hardness, freedom from structure and possible elongation without per- 
manent deformation or change which are met with ina quartz fibre. I must, however, 
defer its discussion for the present or leave it to some one more competent than 
myself. 
4. The Combination of the preceding Three Results. 
The method of combining the results given in the first three sections of this part 
is simple enough. From the first section, the deflection in scale divisions when the 
lead balls are moved from the + to — positions is obtained. Let this be called P. 
The second section gives Q the numerical coefficient of G; thus, Q G is the couple 
exerted upon the fibre. From the third section, the actual couple S that would be 
needed to twist the torsion fibre through an angle of 1 unit (57°3) is obtained 
without any reference to G, and D being the actual distance in units or tenths of a 
scale division from the scale to the mirror measured as explained on p. 17, it follows 
G = PS/4QD. 
The 4 in the denominator is due to the doubling of the angle by reflection and to 
the doubling of the deflection by moving the lead balls from the + to the — positions. 
G is thus obtained in inch?/gramme second* units; to convert it into 
