258 DEVELOPMENT of a certain 



6. It is obvious, that if we had fuppofed A' and B', the firft 

 two coefficients of the feries A' -f- B'cof <p-f C cof 2<p4" &c. 

 known, we could thence have determined A and B the firft two 

 coefficients of the feries A -f- B cof <p + C cof 2<p -f- &c. ; fo that, 

 in either cafe, the remaining coefficients C, D', &c. C, D, &c. 

 which depend upon the firft two in each feries, would alfo be 

 known. 



Hence it follows, that while on the one hand we can pro- 

 ceed from the cafe in which the exponent is 0, to thofe cafes 

 in which the exponents are n — 1, n — 2, &c fo, on the other 

 hand, we can proceed from the fame cafe, to others in which 

 the exponents are n -f- 1, n -j- 2, and thus we may go on, ac- 

 cording to a defcending or afcending fcale, as far as we pleafe. 



7. I now proceed to the chief object of inquiry in this paper, 

 namely, to find convenient geometrical expreffions for the firft two 

 coefficients A and B, in fome particular cafe, where the expo- 

 nent n is the half of an odd number ; and I felect that in which 



n — — |, becaufe of its importance in phyfical aftronomy ; but 



it is evident from what has been already fhewn, that from 



hence we may determine the coefficients, in the cafes of n zz — -, 



&c. and alfo n — — -, n ~ -f- \ t &c. 



Let us therefore affume this equation, 



O 1 + b x — 2ab cof <p) ~ l = A -f B cof <p -f C cof 2^ + &c. ; 



or, putting, for the fake of brevity, g == -, where b is fuppofed 



to be lefs than a, + 



(1 -f- e 2 — 2e cof <p)~ T = a> (A -f B cof <p + C cof 2<p -\- &c.) 

 Then, from what has been already fhewn, (Aft 3.) we have 



va'A 



