OF FLUIDS ON THE MOTION OF PENDULUMS. 



[79] 



Determination of the value of ^//x for water from Coulomb's experiments on the decre- 

 ment of the arc of oscillation of disks, oscillating in their own plane by the force of 

 torsion. 



In correcting the results of the first three experiments for the imperfect elasticity of the 

 wire, Coulomb calculated the values of m given by the four experiments, and subtracted the 

 value given by the fourth from each of the others. But it is at the same time easier and more 

 exact to subtract the value of log (l - m)' 1 given by the fourth experiment from that given by 

 each of the others. For if 



2c 



d0_ 

 dt' 



,dd 



2c — 



dt 



be the moments of two forces, each varying as the velocity, divided by the moment of inertia, 

 the factors by which the initial arc of oscillation must be multiplied to get the arc at the end 

 of the time t, first, when the two forces act together, secondly, when the second force acts 

 alone, are 



respectively, and that, whether the time t be great or small. Hence if we subtract the 

 logarithm of the second factor from that of the first we shall get the logarithm of the 

 factor due to the action of the first force alone. But if we put each factor under the form 

 1 - m, and subtract the m of the second factor from the m of the first, we shall not get the 

 m due to the first force alone, unless t be small enough to allow of our neglecting the squares 

 of ct and c't, or at least the product ct . c't. In truth, when t = r, the quantities m are suffici- 

 ently small to be treated in Coulomb's manner without any material error, since the corrected 

 values of log (l — m), obtained in the two ways, would only differ in the 4th place of 

 decimals. 



The numbers given in the last column of the above table were calculated from the formula 

 (164), on substituting for log (l - m)~ l the numbers found in the first three lines of the 4th 

 column, corrected by subtracting 0-0058. The mean of the three results is 0-05557, but the three 

 experiments are not equally valuable for the determination of y*V. For the three numbers 

 from which ^/fi was deduced are 0-0510, 0*0152, 0*0077, and a given error in the first of these 

 numbers would produce a smaller error in ^/ lk than that which would be produced by the 

 same error in the second, still more, than that which would be produced by the same error in 

 the third. If we multiply the three values of -y//u' by 510, 152, and 77, respectively, and 



