RECTIFICATION of the ELLIPSIS, &c. I8i 



(p{a^-{-I)^ — 2abco{(p) r: 2" X 

 f <p (fin|-)/" becaufe z (finf) = i — cof <p : we thus obtain 



/■ / (ft ^^ 



IT 



the whole fluent to be taken when (p^ in jr, or - ^ 



2 2 



2 



If we put x zzzTm -, we fhall have 



2" >< /tT^F :z I + «-' + /3^ + 7^ + &c, 



;w 



the whole fluent to be taken when .v = i ; and in this formula 

 n is any number fradlional or integral, pofitive or negative ; and 

 a, (B, 7, &c. the coefficients of the binomial raifed to a power 

 of which the exponent is n. 



When « is a whole pofitive number, 



f— - — — ^-i i^ ••-, m the cale when ^ =: i : 



J^l—X^ 2.4.6 2Tl 2' 



And fo, 2" x ^-'-^----^'"~'^ - I + «^ + iS^ + r^ + &c. 



' 2. 4. 6 .... 2a 



Now, 2" X ' '•3- 5 • • -C^" — I J jg jjQ Other than the coefficient of 



2. 4. 6 ... . in 



the middle term of a binomial, raifed to the power exprelTed by 

 2« : Hence we have a very curious property of thofe numbers: 

 viz. that the/urn of the fquares of the coefficients of a binomial, the 

 exponent being n, is equal to the coefficient of the middle term of a 

 binomial, of which the exponent is 2«. 



Another remark, which I have to offer on this fubjedl, may 

 be confidered not only as ctirious, in an analytical point df view, 

 but as, in fome meafure, accomplifliing an objedt that has much 

 engaged the attention of mathematicians. 



In the computation of the planetary difturbances, it becomes 



neceffary to evolve the fradion {a"- -{■ b^ — 2ab co£<p) into a. 



feries 



