XI. A New Method of exprejfing the Coefficients of 

 the Development of the Algebraic Formula 

 (a* + ^'- — zah co{<pf, 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 /or the Rectification of any Arch 

 of an Ellipse. By Mr WILLIAM Wallace, Afftjlant- 

 T'eacber of the Mathematics in the Academy of Perth. 



IRead Jan. ix. 1802.] 



I . T N calculating the efl:e£t of the mutual adlion of two pla 

 JL nets upon each other, it has' been found neceffary to 

 develop the algebi'aic formula {a- -{- (^' — • 2ab cof (p)" into a 

 feries of this form, A + Bcof(p+ C cof 2(p + D cof 3(p+ &c. 

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

 (p denotes the angle of commutation.; and the values of w, more 



immediately the fubjedl of confideration, 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 . firfl applied to the folution of 

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

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

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

 could be exprelFed 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 



