p — I 
n % z s 
• [ 312 ] 
. ?— 1 , tzi. "~ 2 v 
-r -6- + p* —p~p x 6 
-j- Gfc. The hyperbolic logarithm of th e fecond 
ratio is the feries q — - — — X n z — q —— r (- 
q P q 
IH1 _L till _ 1H- 2 x + &c. It will ap- 
q 4 1 /> + p~ q z 4 
pear from examining thefe two feriefes (y all along 
fuppofed greater than p) that while z is fmall the 
value of each of them is pofitive, and increafes as z 
increafes till it becomes a maximum , after which it 
decreafes till it becomes nothing, and after that ne- 
gative ; which demonftrates this article. 
15. The former of the two ratios in Art. 13. (q be- 
ing greater than p') is at . firft, while z is increafing 
from nothing, lefs than the fecond ratio j and does 
not become equal to it, till fome time after both 
ratios have been the greateft pofiible. 
Upon confidering the two feriefes in the laft Art. it 
will appear that the firft term of the firft feries is always 
pofitive, the fecond negative, the third alfo negative, 
after which the terms become alternately pofitive and 
negative. On the other hand, it will appear that in the 
fecond feries the two firft terms are always pofitive, 
and all that follow negative. But as the feriefes con- 
verge very faft when z is fmall, the fecond term 
being negative in the firft leries and pofitive in the 
fecond, has a greater effedt in making the firfi; feries- 
lefs than the fecond, than can be compenfated for by 
the terms being afterwards alternately negative and 
j , pofitive 
