for the Development of the Function f (a + x.) £13 



This equation can be easily put under the form {a + x) 



n+n X ,,.'"' n.n—\ .r* / , \" , 



— a {a + x) -—— - — ; — Tj- (a + x) + 



\ a + X ^ 1- 2 (a +.v)* 



— — (a + .r) — &c. and thence we 



1-2-3 (a + X*) ^ 



calculate (a + a;) . ,(i - -• -_^^, + -p^- ^^qj^, - 



+ &c. = a ; or,bv avervsimple 



1- 2-3 (a + .()•' ^ ^ J ^ J - r 



re 

 transposition, we arrive at the following result : «_ (a + x) 



— n n X n.n — I .x* n. n—\. n — 2 . 



~^ ~ I a + X "^ 1-2 (a -f- .%■)'» F2- 3 



This expression, were it only for the singular manner in 

 which we have obtained it, is certainly worthy of attention ; 

 but besides this, it evidently furnishes a more convergent 

 series for the extraction of roots, than that which is procured 

 from the ordinary development of the binomial. For if we 



1 • 1 / \" ^* /■ '^ ■'^' 



change n mto — n, we have (a + ^j = " I ^ + , — ; — 



n.n + I A* n.n + 1. n + 2. .r^ , . , 



H 1- •; "5 + &c. and, 



1-2 (a + *)« r 2- 3 (a + .v/ 



putting X = 1, the form of the equation is changed into 



n nl n n. n + I n.n-\-\.n-\-1 



(a + ]) ^ a.[x+ __ + _ -^-^_+ -^.^r^---y 



+ &c.) where the scries is always the znore convergent as 



a is the larger. 



We shall next apply the general theorem :p [a + x) = 



X d. f {a + x) .\* d.^ <p {a + x) . ^ , 



f (a) + ?L-' :L_^_ IIA. L 4. SiC. to the 



^ '' -^ 1 du 1-2 du-^ 



determination of the sine and cosine of a + x; anj in the 



first instance making 9 (« 4- a; = Sin. a + Xy we have 



__J:_\ -T-Z _ Cos. a + A-: — ^; — - — - — — Sm. a + x- 

 du d u- 



d* a la -\- k) - .... 



~' l\ = — Cos. {a + x); 5cc.; and sub^itu'Dig 



thssp 

 O 3 



