178 Proceedings of the Royal Society of Edinburgh. [Sess. 
where b and t 0 are constants, and n = m— 2. This is identical with the 
log (x + a), the points can in many cases, such as brass at the ordinary 
temperature, be made to lie along one straight line, when a suitable value 
of a has been chosen. The tangent of this line’s inclination to the log y 
axis then gives the value of n. In other cases, however, such as copper 
in the soft condition, it was found ( Proc . Roy. Soc. Edin., p. 436, 1910-11) 
that no one value of a could be found to cause the points over the range 
taken to fall on one straight line, but that an s-shaped curve was in 
general got. This required two separate values of a for large and small 
oscillations respectively, and two straight lines, inclined at an angle concave 
to the origin, were obtained. 
In order to test the applicability of the empirical law further, a fuller 
approximation to the true representation of results was taken as follows : 
If 6 i and 0 2 be the ranges of the first and second out swings respect- 
ively, the loss of energy is represented by 
and this loss will, taking the arithmetic mean of the two swings, be more 
approximately equal to 
the interval of time between the swings being taken as unity. Or, sub- 
stituting the proportional scale readings, 
equation used, and it has been shown that if log y be plotted against 
instead of pQ m as formerly. 
Thus we can write 
But if we suppose 
= y-i - d V \ » 
then 
Hty\ ~ dy l )dy 1 =p[y 1 m + (y 1 - dyj™] 
Approximately, p being small, 
k(2y - dy)dy = p[2 y m - my m ~ 1 dy\. 
Hence 
