gives complete tables of the same ratio. However, given T, and 
H and a table of ordinary hyperbolic functions, it is possible to 
find the above length without recourse to these graphs and tables. 
Equation (12.5) is equation (12.4) written down again. Substi- 
tute the expression for 27/L on the right in (12.5) for the 27r/L 
under the hyperbolic cotangent on the right of equation (12.4). The 
result is equation (12.6). Again substitute the value of 27/L in 
(12.5) into the far right of (12.6). The result is equation (12.7). 
Do it again. The result is equation (12.8). After an infinite 
number of substitutions the result is that 27/L is given as a function 
of # and H* alone on the right hand side of the equation. Thus, in 
a sense, equation (12.4) has been solved for 2r/L in terms of H and 
pe The new function suggested by equation (12.9) is defined by 
equation (12.10) to be the Itcoth of Hu-/g, [or (Hp @/g)]. The 
symbol, Itcoth(Hp “/z), is to be read as the iterated hyperbolic 
cotangent of Hy /e. It can also be pronounced easily just as it 
reads. The Itcoth appears to be a brand new function, never written 
down before. 
The point of the new function is that substitution of Hp °/g 
for H2r/L at the far right in the iteration makes no error in the 
value of the function. In fact only seven or eight iterations yield 
three place accuracy for the Itcoth starting out with Hi /e instead 
of H2r/L if Hp °/g is fairly large. Near zero values, many more 
iterations are needed. 
Table 17 illustrates this point. Let the depth be one eighth 
*The usual notation for this symbol is h, but H is used here in 
order to avoid confusion with the h of Chapter 10. 
30 
