174 REPORT— 1898. 



Semi-logarithmic Coordinates. 



40. For some purposes it would be an advantage to plot the logarithm 

 of one variable against the real value of the other. For instance, suppose 

 it were desired to find the logarithmic decrement of an amplitude which 

 we Avere investigating. This could be done by setting out the logarithms 

 of swings against the number of the vibration. If the logarithmic decre- 

 ment did not vary the curve would be a straight line, and the logarithmic 

 decrement would be proportional to the tangent of the angle which this 

 line makes with the decreasing direction of the axis of the number of the 

 swing. If, however, the logarithmic decrement varied with the time, the 

 tangent to the curve at any point would give the logarithmic decrement 

 at that instant. This example is only onfe of many which suggest them- 

 selves ; the usefulness of the method in practical work would chiefly 

 depend on the fact that it is easier to ' smooth out ' a sei'ies of points 

 which should be in a straight line than it would be to draw a logarithmic 

 curve with an elastic rod, as would have to be done if the results were 

 plotted on ordinary square paper. 



The Graphical Computation of the Hyperbolic Functions. 



41. By the use of semi-logarithmic coordinates it is easy to construct 

 a chart which shall give the values of the six usually used hyperbolic 

 functions for a given value of the independent variable. I have drawn a 

 diagram of this sort which has an accuracy of about one in two hundred, 

 and hope to be able to publish one on a larger scale which shall have a 

 greater accuracy. 



For the sake of clearness let the logarithmic scale be regarded as the 

 vertical one and the scale of equal parts as horizontal ; further, let the 

 component logarithmic scales in the vertical scale be equal to a unit of 

 length on the horizontal scale. 



42. The Meaning of a Straight Line on the Chart. — If a straight line 

 be drawn arbitrarily on such a chart (see fig. 5) it gives a series of values 

 for %v andysuch that — 



/=10™"c=A;"c, 



where /is the reading on the logarithmic scale and ?t that on the horizontal 

 scale ; c is the reading on the logarithmic scale of the point of intersection 

 of the line with the vertical line u — o ; k is the tangent of the angle which 

 the line makes with the increasing direction of the axis of ?(. 

 Thus any curve having an exponential equation of the form 



is traced at once by drawing a straight line on the semi-logarithmic chart. 

 No numerical computation is necessary ; the value of c is mg-rked on the 

 vertical scale when u=o ; if k be then marked on the vertical line 

 through tt=l, a line joining this last point to the point /=1, m=o (which 

 may be called the origin) gives us the graph of 



f=h\ 



A line parallel to this through the first marked point gives the 

 graph of 



f=ch\ 



