ALGEBRAIC FORMULA. 26s 



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

 &c have to each other. 



For, e denoting the excentricity of the ellipfe (E), and * that 



of the ellipfe (E y ), we have found e % — - , *' . T , (Art. 8.) there- 

 fore, v/i — e^ — ~~*, and confequently s — |",^ , fo that 



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

 e' for g ; and, fince we have the two equations 



(i + f)E-(2 + r)E + f (i+^)E'-o, 

 (1 +e') E— (2 + c')E'+S (1 +*') 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 unfavourable 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 practical rules, 

 the fubftance of the preceding inveftigations for determining 

 the firft two coefficients of the development of the formula 



{a 1 -f- b~ — 2ab cof <p) n , in the cafe when n — — |; but from thefe, 



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

 eafy to determine any number of the coefficients, when 11 is the 

 half of any whole number, pofitive or negative. 

 To compute A and B in the equation 



{a"~ + b z — 2ab cof. (p)~ * =z A -f B cof <p -f C cof 2<p -f &c. 



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



tranfverfe axis pi 1, the femi- conjugate = ^j, and therefore, 



1 • • 2l/ (7/7 



the excentricity ~ — 1 — . 



2. Also 



