VIII. A New Series for the Rectification of the Ellipsis ; 

 together with fome Observations on the Evolution 

 of the Formula {O" -{- h^ — 2a h cofcp)^ By James 

 Ivory, A. M. Communicated by John Tlatfair^ Pro- 

 fejfor of Mathematics in the Univerjity of Edinburgh, 



[Read Nov. 7. 1796.] 



Dear Sir, 



HAVING, as you know, beftowed a good deal of time and 

 attention on the fludy of that part of phyfical aftronomy 

 which relates to the mutual diflurbances of the planets, I have, 

 naturally, been led to confider the various methods of refolving 

 the formula (a"" -f- ^* — 2ab coftp)" into infinite feries of the 

 form A + Bcof(p + Ccof2<p+ &c. In the courfe of thefe in- 

 veftigations, a feries for the redlification of the ellipfis occurred 

 to me, remarkable for its limplicity, as well as its rapid conver- 

 gency. As I believe it to be new, 1 fend it you, inclofed, to- 

 gether with fome remarks on the evolution of the formula juft 

 mentioned, which, if you think proper, you may fubmit to the 

 confideration of the Royal Society. 



I am. Dear Sir, 



Your*s, &c. 



DouGLASTOWN, nearforfar,") TaMES IVORY. 



20th 0£t(Jber 1796. 3 •' 



17o Mr John Playfair, Pro- 

 fejfor of Mathematics, Is'c, 



Vol. IV. Y Let 



