[90] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION 



or 16 lines. In the case of the short pendulum, the differences in 8 cases out of 10 had the 

 same sign as before. The mean difference was 0.025, and the mean error 0.043. The arcs of 

 oscillation were nearly the same as before ; but inasmuch as the axis of suspension was nearer 

 to the scale than before, the initial value of n was only about 12 or 13 lines, and the final 

 value about 7 lines. When the results of some of the experiments were laid down on paper, by 

 abscissas taken proportional to the times and ordinates to the logarithms of /u, it was found that 

 in the case of the long pendulum the line so drawn was decidedly curved, the concavity being 

 turned toward the side of the positive ordinates. The curvature of the line belonging to 

 the short pendulum could hardly be made out, or at least separated from the effects of 

 errors of observation. The experiments 9, 10, 11, having been treated numerically in the 

 same way as the experiments 1 — 5, led to much the same result. In the 16 experiments with 

 the ivory sphere and short pendulum contained in the experiments Nos. 12, 13, 14, and 15, 

 the excess of the calculated over the observed value of fi was more apparent, the mean 

 excess amounting to 0.129. The reason of this probably was, that the observations with the 

 ivory sphere were made through a somewhat wider range of arc than those with the brass 

 sphere. 



It appears then that at least in the case of the long pendulum a correction is necessary, in 

 order to clear the observed decrease in the arc of oscillation from the effect of that part of the 

 resistance which increases with the arc more rapidly than if it varied as the first power of the 

 velocity, and so to reduce the observed rate of decrease to what would have been observed in 

 the case of indefinitely small oscillations. 



75. In Coulomb's experiments it appeared that the resistance was composed of two terms, 

 one involving the first power, and the other the square of the velocity. If we suppose the 

 same law to hold good in the present case, and denote the amplitude of oscillation at the end 

 of the time t, measured as an angle, by a, we shall obtain 



Tf- A «-**> • • • 073) 



where A and B are certain constants. We must now endeavour to obtain A from the results 

 of observation. Since the substitution for a of a quantity proportional to a will only change 

 the constant B in (173), and the numerical value of this constant is not required for com- 

 parison with theory, we may substitute for a the number of lines read off on the scale as 

 entered in Bessel's tables in the columns headed p. 



I have employed four different methods to obtain A from the observed results. The 

 one I am about to give is the shortest of the four, and is sufficiently accurate for the 

 purpose. 



The equation (173) gives after dividing by a 



d log a , _ 



-g A- Ba (174) 



