ALGEBRAIC FORMULA. 267 



14. There are two feries, which, fo far as I know, are the bed 

 adapted to the reflification of the ellipfe, of any hitherto pubhfh- 

 ed ; becavife, while both are well fuited to a moderate degree of 

 cxcentrkity, the one converges with great rapidity when the 

 excentricity is confiderable, and the other when it is fmall. 



The firftof thefe feries, which appears to have been origi- 

 nally given by Mr Ecler *, converges by the powers of the fe- 

 mi-conjugate axis, and may be exprefled, fo as to exhibit the 

 law of the feries, as follows : 



Let the femi-tranfverfe axis rz r, 

 the femi-conjugate axis n c. 



Then, - of the perimeter of the ellipfe is equal to 



i + C<^ +i'2^' + i:6-2~^ +4-:o-5:^^ +&c.)hyp. iog.7-- 



4 



_3.i,4n L_) 



. 4 a \iz 3.3.4/ 



. _3J.»J^«(i3 1 i_\ 



4.6 2.4 ^20 2.3.4 4.5.6/ 



{}9_ I I L_"\ 



\28 2.3.4 4.q.6 6.7.8/ 



LH.LH^s . 



4.6.8 2.4.6 V28 2.3.4 4-J.6 6.7. 



— &c. 



The other feries was given by Mr Ivory, in a very ingenious 

 paper upon the fame fubjedl we have been juft now confider- 

 ing f . It converges by the difference of the axes, divided by 

 their fum ; fo that if we fuppofe the femi-axes, as before, to be 



denoted by i and £■; and put d z=. f- r^, and sr for 3.14T59, &c. 

 Half the perimeter 



* In a traft entitled " AnimadverCones in Reaificatlonftn Ellipfis," whicU 

 forms part of the fecond volume of his Opufcula. 



t Tranfaftions of the Royal Society of Edinburgh, vol. iv. 



