QUADRATURE of the CONIC SECTIONS, &c. 311 

 here 



R > - T m +i) 



3 



but R < _ T(w + 1) X T(» + . 



T( m) — T(« + i) 



As thefe limits to R differ but little when the terms T( m ) y 

 T(m +t ) are confiderably advanced in the feries, the latter may 

 be exprefled more conveniently for calculation thus 



I (4TO+O T(m))T(«+i) 



R < - T(« +I ) + — rFFT — t - "^ * 



43. Let us next inveftigate a feries for the quadrature of the 

 hyperbola, which may be analogous to our fecond leries for the 

 rectification of the circle. For this purpofe, proceeding as at 



Art. 1 3. we refume the formula ? zr -4- - tan ~ S, 



J tan S 2 tan ~ S 2 



and taking the fquare of each fide of the equation, get 



_ I 



+ I tan 2 - 1 S + I. 



tan 2 S " " 4 tan 2 i S 4 2 2' 



Inftead 



be deduced one from another by the help of the common trigonometrical tables. 

 It is this, 



7 = TSS7 ~ < W + 1 fiD *"+ -4 W " + I fi»«"- + T W ■+ T„ + 1) + R). 



The arches a,', a", a'", « ,v , &c. are to be deduced one from another as follows. 



Take a fuch that fin a — — - — , then, fina'~ tan - a, fin a" — tan I a' fin a'" ~~" 



ab j 2 2 "~" 



tani a", fin a iv =r tan -*'", See. The fymbols T(»), T(» + j) and R, denote, 

 the fame things as in the other form of the feries. 



