ox THE USE OF LOGARITHMIC COORDINATES. 175 



A line through the origin and the point f^\0, ii=.\, gives 



/=10", 



and this line (which is at 45° to axis of m, since log 10^1 by hypo- 

 thesis) is a table of common logarithms ; the readings of the u scale are 

 the logarithms of the readings of the _/ scale. 

 For a line representing 



the ti is in any case the logarithmic of f to the base k, so that the semi- 

 logarithmic chart constitutes an infinite series of logarithmic tables to 

 any base. 



43. To draw a line representing e". This line passes through the 

 origin and the point f=^e, i(,^=\. The e" line is a table of natural 

 logarithms, .u being the natural logarithmic of f. 



The accuracy with which such graphic tabulation may be performed 

 is limited by the size and accuracy of the graduations. The size of the 

 16 graduations is a matter of choice, as semi-logarithmic charts have to 

 be made and cannot be bought. The logarithmic scales are only obtain- 

 able in one size on the ruled paper, but other logarithmic scales can be 

 obtained from slide rules. 



Semi-logarithmic charts are free from the defect of ordinary squared 

 paper (as regaixls the proportionate accuracy of plotting) in the vertical 

 direction. The percentage accuracy of the horizontal scale varies as the 

 distance from zero of the abscissa. 



44. To proceed to compute the hyperbolic functions other straight 

 lines must be ruled on the chart. These are inserted without computa- 

 tion. The line representing 



(■"■ 



f= 



' 2 



is parallel to the e" line, and is drawn through the point f=-'b, u=0. 

 The reflexions of these lines in the liney=l give us the e"" and 2e~" lines, 

 while a line drawn through y='5, ic=0, parallel to the last two, gives 

 . e-" 

 •^ 2 



45. The Semi-lorfarithmic Graph of Sinh u. — Any point on this curve 



is found by reading off the values of — and — from the appropriate lines 



for a chosen value of n. Subtract the latter from the former numerically 

 and mark the result on the vertical scale through u. In this way the 

 curve can be readily drawn. 



46. The Cosh u Curve. — This is conveniently drawn at the same time 



as the sinh v, curve, the values being added. Both curves are asymptotic 



e" 

 to the — line. 



47. The Graphs of Cosech u and Sech u. — These functions being the 

 reciprocals of those just traced, we have only to make the diagram 

 symmetrical about the line /— 1 to obtain the two new curves. They 

 are asymptotic to the line 2e"". 



48. Tlie Grajihs of Tanh u and Coth u. — These are drawn in from the 

 sinh 10 and cosh iv curves ; the vertical distances between these curves 



