or ELECTEIC SIGNALS THEOEGH SUBMAEINE CABLES. 
1017 
Quite similarly, if a be known, the number of beats per minute corresponding to any 
amplitude could be determined ; but, practically, a need not enter into the calculation 
when treating of any given cable. The speed, amplitude, and length are here the three 
elements of every problem, and when two of these are known the third can be deter- 
mined ; but here it may be observed, that as the speed, multiplied into a, is constant for 
each amplitude, so will the speed, multiplied into the square of the length, be constant 
for each amplitude, and the scale of abscissae may be so chosen as for any one cable to 
give directly this product by simple inspection. 
It is this scale for the Eed Sea cable which is drawn at the foot of the curve, fig. 11, and 
which enables the number of dots corresponding to every amplitude to be ascertained 
directly, and it is by this scale that the dots, crosses, or circles from Table XV. are put 
on the figure. 
When, as in the present paper, the speed is taken as twice the number of dots, and 
the unit length is one knot, the ratio of the two scales must clearly be such that if 
d =the number of divisions in the upper scale, 
D=the corresponding number on the lower scale, 
L=the length in knots, 
then 2dL? 
a — P • 
Thus, taking a length of 1000 knots and a speed of 100 dots per minute, D=2xl0®, 
d=12"6, and hence a=0T26 ; and the same value would be obtained whatever number 
of dots had been chosen. 
This may be looked on as the mean value of a determined from twenty observations, 
since this ratio of the scales brought all the various circles, crosses, and stars into the 
closest approximation with the curve. The values of a for any other length are 
inversely proportional to the square of the lengths. 
The algebraic headings of the different columns will allow them to be still further 
extended by those who may require to use the Table, as they virtually contain the 
equation of the curve. 
