RECTIFICATION of the ELLIPSIS, &c, iSi 



/n 



J (p (finy), becaufe 2 (finl-) = i — cof <p : we thus obtain 



the whole fluent to be taken when cp- rz :z?, or |- 

 If we put X ~ fin -, we fliall have 



2 i~ 



^J'yT^T^ - I + a"" + /S^ + 7^ 4- &c, 



the whole fluent to be taken when Af rr i ; and in this formula 

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

 a, /3, 7, &c. the coefiicients of the binomial raifed to a power 

 of which the exponent is n. 



When « is a whole pofitive number, 



f- — n — ^-? ^^ -•-, m the cale when x ■=! i i 



Jy/X—X' 2.4.6 2« a' 



And fo, 2^^ X i-.^5.---(^^-r> --. I ^ ^a ^ /3^ _^ . ^ S^c. 



' a.4.6.... 2« ' I r- 1 / I 



Now, 2'" X ^•3-5 • --(^a — I ) .g ^^ other than the coefiicient of 



2. 4. 6 .... 2« 



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

 in : Hence we have a very curious property of thofe numbers; 

 viz. that the fum 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 2n. 



Another remark, which I have to ofi^er on this fubjedl, may ■ 

 be confidered not only as curious, in an analytical point of view, 

 but as, in fome meafure, accomplifliing an objed: that has much 

 engaged the attention of mathematicians. 



In the computation of the planetary diflurbances, it becomes 



neceflary to evolve the fraction {a'^ -\- b"^ — 2ab cof ^) into a 



feries 



