28o DEVELOPMENT of a oefValn 



^..^ The quantities ^'k.^*, /"i, &c; approach veryfaftto o, zxA the 

 arches 0', (p", «^"', &C. approach to, a jCertam.Umit, which .let . us 



denote by ^. ■ -^ r j ■ 



CoMP UTE alfo thefe three quantitie^, '^" '■'-'"^v' ' ^^^» 



The elliptlci arch W = ^^i-.Q:)+^K. '' ^-^-, o.faitni «., 

 When, <p, becomes' a quadrant,, th? 'fi^esof i(pV4?^^ Sip'", &c. 

 ■ire evidently eacTi'bV'fo that/putjirig^ 



in this cafe, — ^, and, putting E^^q 4^i>9te. the, .whple.'.elliiPtic 

 quadrant, we have ,eixr. olr 



^/ E =:f p(i — ^Q7.)ioriinD3:<i orb ^ 



No"w it is worthy of remark, that ' the expTeffion Pp:'^^l?Q^) 



is common to the indefinite arch z attd'the whole quadrant E ; . 



■hence it follows that the indefinite arch may be alfo exprelTed, 

 , — -uinoo -jii. 



thus, y '• 



i.2. From thi3,laft formula \ye n>ay deriye manyof thofe cu- 

 riovis relations 'which are known to fubfift, between certain af- 

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

 whole elHptic quadrant *. For .example, w^ may hence deduce 

 that very remarkable property of the ellipfe, whichs^is. cpmrnon- 

 ly known by -the >name of Task ami's theorem,.namay, 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 cpuflrudion, 

 , ,^-=-- - "'<5>3 flfi . ir^;^-^^, ■ fuch 



• The properties of the ellipf&lieEe aUudedi'^o have been explained by EuLEK, 

 and fomrif them have 'alfo been obferVed by Landen. 



i'T 



aM 



