176 



REPORT — 1898. 



when set out above f=\ give the coth tt curve, and when set out below, 

 the tanh ?t curve.' 



49. A figure constructed thus forms a check on more elaborate 

 numerical computation. It should be noticed that it can be drawn almost 

 entirely by graphical methods, the only numerical value required being 

 the number e. 



It is obvious that the semi-logarithmic chart provides a short and easy 

 method of calculating the cooi-dinates of catenaries of which cosh u (on 

 squared paper) is one. 



On some Points in the Geometry of the Semi-logai-ithmic Graphs of the 



Hyperbolic Functions. 



50. If we write down the other hyperbolic functions of m when 

 sinh M=rt we find that the values of the six functions are identical with 



Fig. G. 



those of u', whei'e it' is defined by sinh %'=-. The six equations thus 



a 



obtained may be expressed by saying that any function of a is equal to that 



function of a' which stands opposite to it in the table — 



sinh cosech 



cosh coth 



tanh sech. 



Thus (see fig. 6) if a straight line such as AA' be drawn perpendicular 

 to the axis of u and C be the intersection of this line with any of the six 

 curves (in the figure with coth u) ; then if a straight line be drawn 

 parallel to the axis of u to cut the curve indicated in above table at the 

 point c (in this case cosh ?t) a vertical straight line through c will cut all 



When u is small, put tanh u=u and coth u = . 



u 



