2 8o DEVELOPMENT of a certain 



The quantities f'jrfoi <?'", &c. approach very fad to q, and the 



arches <£>', <p\ <£•'", Sue. approach to a certain limit, which let us 



■ . - 



denote by 8. 



Compute alfo thefe three quantities, 

 P = (i-f-0(i+0(i+O&c. 



X^" 2~ 2.2 ' 2.2.2 ' 2.2.2.2. '" > 



R^ -^frW- K . 2 .2 V 2 rin 4^ +-2 - 2 r i t rr rrt " fin8 *> + &c> 



The elliptic arch z = flP(i _*QJ+/R. 



When <p .becomes a quadrant,, the' fines of 2<p\ £$>", 8 <£>"*, &c 

 are evidently each o, fo that, putting V '= 3. 1-4159, &c. we have 



in this cafe, 6 =yg s and, putting E Jo denote, the whole elliptic 



*. 

 quadrant, we have 



Now it is worthy of remark, that : the expremon P(r — <?Q^) 

 is common to the indefinite arch z and the whole quadrant E ; 

 hence it follows that the indefinite arch may be alfo exprefTed 

 thus, 



z = — E + e B-. 



12. From thislaft formula we may derive many of thofe cu- 

 nous relations which are known to fubilft. between certain af- 

 fignable elliptic arches, as alfo between, thefe arches and the 

 whole elliptic quadrant *, For example, we may hence deduce 

 that very remarkable property of the ellipfe, which, is common- 

 ly known by the name ofTAGNANi's theorem, namely, that an 

 elliptic arch, reckoned from the extremity of either axis being 

 fuppofed given, another arch, reckoned from the extremity of 

 the other axis, may be found, by a geometrical conflru&ion, 



f uch 



* The properties of the ellipfe here alluded ffo have been explained byEuLER, 

 and fome of them have alfo be£n obferved by Landen. 



