ALGEBRAIC FORMULA. 267 



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

 adapted to the rectification of the ellipfe, of any hitherto publifh- 

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

 cxcentricity, the one converges with great rapidity when the 

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



The firft of thefe feries, which appears to have been origi- 

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

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

 law of the feries, as follows : 



Let the femi-tranfverfe axis zz 1, 

 the femi-conjugate axis — c. 



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



_3.i,,(.l__!_) 



_ 3J . ^(LS 1 i\ 



4.6 2.4 ^20 2.3.4 4-5.6/ 



3.5.7 1.3.5 s (12.- l , _i I \ 



4.6.8 ' 2.4.6 c V28 "2.3.4 4.5.6 6.7.8/ 



— &c. 



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

 paper upon the fame fubject 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 1 and c ; and put d — ^~ c , and <z for 3-14*59, &c. 



Half the perimeter 



I 



* In a tracl: entitled " Animadverfiones in Re&ificationftn Ellipfis," which 

 forms part of the fecond volume of his Opufcula. 



f Tranfa&ions of the Royal Society of Edinburgh, vol. iv. 



