ALGEBRAIC FORMULA. 25^ 



tiiemfelves, that any two adjoining ellipfes in the feries E\ E, E', 

 &c. have to each other. 



For, e denoting the excentricity of the elHpre (E), and t that 

 of the elHpfe (E'), we have found ^^ = (7^^, (Art. 8.) there-' 



fore, K/i—e-'^zz i^, and confequently 6 z= \ ~^^^ , fothat 



we may change the fymbols (E) and (E') for E and E', and put 

 ^ for t ; and, fince we have the two equations 



(i + 0F-(2 + 0E + c(i+^)E' = o, 

 (i +0 E— (2 + O E' + c' (I + c') E" = o, 

 it will immediately follow, that if any of the two ellipfes E and 

 E', (by which we have exprefled the coefficients A, B, &c.) have 

 fuch a degree of excentricity, as to be imfavourable to numerical 

 calculation, we can exprefs that ellipfe by means of the other 

 one, and a third ellipfe, which may be more or lefs excentric 

 than either of the other two ellipfes, juft as we pleafe. 



13. I SHALL now give, in the form of pradlical rules, 

 the fubftance of the preceding invefligations for determining 

 the firft two coefficients of the development of the formula 

 («■' -\-h- — 2ab cof (p)", in the cafe when n zz — 2 j but f^ora thefe 



and from what has been delivered in Articles 4. and 5.-, it is 

 eafy to determine any number of the coefficients, when n is the 

 half of any whole number, pofitive or negative. 

 To compute A and B in the equation 

 {a^ -\-b^ — 2ab cof <?)—'- =: A + B cof <^ + C cof 2^ + &c. 

 I. Find E half the perimeter of an ellipfe, of which the femi- 

 tranfverfe axis =; 1/ thte' femi-conjugate = J=|, and therefore, 

 the excentricity zz i '"''. 



2. Also 



