318 NEW SERIES for the 
being greater than the former, but lefs than the latter. This 
laft limit may alfo be otherwife expreffed thus, 
(16 Timt1) — T m) Tent1) 
15 (T(m) ne 
= T(m4r1) + 
Or compute the feries of HARRIS t, t', t’, &c. one from 
another by thefe formulz 
Jabra pT? | raver 
~ absae dr PNG RE pee 
Then fhall 
Ln BS a8 82 Eo ns 
Poker eae seal! + Bt teh tb Tob Tot TR), 
the fymbols Tym), T(m+1), and R, being put to denote the 
fame as before. 
49. WE might now inveftigate other feries for the quadra- 
ture of an hyperbolic feétor, fimilar to the third and fourth fe- 
ries we have found for the rectification of an arch of a circle; 
but this inquiry would extend the Paper to too great a length. 
For this reafon, and alfo becaufe the manner of proceeding in 
the one cafe is exactly the fame as has been followed in the 
other, it feems unneceflary, in the cafe of the hyperbola, to ex- 
tend our inquiries farther. I fhall therefore now proceed to 
the third and laft object propofed in this Paper, namely, the in- 
veftigation of formule for the calculation of logarithms, be- 
ginning with a few remarks that may ferve to connect thefe 
formule with the common theory. 
50. Ir is ufually fhewn by writers on this fubject, that all 
numbers whatever are confidered as equal, or nearly equal, to 
one 
