258 DEVELOPMENT' of a certain 



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

 two coefficients of the feries A' + B'cof ip-f C cof 2(p + &c. 

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

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

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

 which depend upon the fird 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 «, to thofe cafes 

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

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

 the exponents are n-f i, « + 2, and thus we may go on, ac- 

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



y. I NOW proceed to the chief objeifl of inquiry in this paper, 

 namely, to find convenient geometrical expreffions for the firfl two 

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

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



« z= .— -, 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 « = — -, 



&c. and alfo n = — h " — "^l' ^^' 



Let lis therefore alTume this equation, 



(«• + b' — 2ab cof (p) ~^ = A -f B cof (p + C cof 2(p + &c. ; 

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

 to be lefs than <?, 



(i ^ e^ _ 2£ cof ?>)~^' =: a^ (A -f B cof (p -f C cof 2(p -f &c.) 

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



