XI. A New Method of expreffing the Coefficients of 

 the Development of the Algebraic Formula 

 (a x -{-b l — 2abco£<p) n , by *means of the Perimeters of two 

 Ellipfes, when n denotes the Half of any Odd Number ; to 

 gether with an Appendix, containing the Investiga- 

 tion of a Formula/^ the Rectification of any Arch 

 of an Ellipse. By Mr WILLIAM WALLACE, Affijlant- 

 7'eacber of the Mathematics in the Academy of Perth. 



{Read Jan. n. 1802.] 



I. IN calculating the effect of the mutual action of two pla> 

 X nets upon each other, it has been found neceflary to 

 develop the algebraic formula (a 1 -f- b z — 2ab cof <p) n into a 

 feries of this form, A-f- B cof <p-f- C cof 2<p -f- D cof 3<p -J- &c. 

 Here a and b denote the diftances of the planets from the fun ; 

 <p denotes the angle of commutation ; and the values of », more 



immediately the fnbject of confederation, are — -, and — — . 



The determination of the coefficients A, B, C, &c. in thefe 

 cafes, appears to have been confidered as a matter of difficulty 

 by the mathematicians who firft applied to the folution of 

 the problem ; for they found, that although it was only ne- 

 ceflary to compute the firft two coefficients A and B, the reft 

 being eafily derived from them, yet it did not appear that they 

 could be exprefTed in finite terms, nor even by means of circu- 

 lar arches, or by logarithms. Recourfe was therefore had to 

 other methods, and chiefly to the method of infinite feries ; but 



Vol. V.— P. II. K k as 



