108

Lastly from the equation y — ~j we havey= which, by substituting s/z^+c 1 for t,becomes y =^==.; or, by taking the cor-rect fluent of each side (observing that y andz begin together) y — c. hyp, log.

which, by subst. = c. hyp, log. v/ 2 cx

=c.hyp.log.^= c.hyp.log. (tang. <£--[-sec. < p)=c.hyp.log. tang. (45°-|-^f>).

Or since, by nature of logarithms, thehyperbolic logarithm of any number is equalto 2.302585 multiplied by the commontabular logarithm of that number ; by put-ting 2.302585 = m and the word log. toexpress the common tabular logarithm, theforegoing equations may be transformed

into the following, viz. 3/= cm log. z +c

_ 1 ™ 2 CX X -^r~C'\~X "I

= cm log. —= cm log. =

cm log. (tang. <p -f sec. ?) = cm log. tang.(45°—}— £<p). But it is to be observed, thatin taking the logarithms from a table of lo-garithmic sines, tangents, secants, &c., 10 must