212 Some Observations on Dr. Taylor's Theorem 



about to enter will explain a method by which the one 



theorem may be deduced from the other; and at the same 



time it will illustr.ate the analytical distinction betwixt them. 



By the theorem of Taylor we have <p {a -^ x) =■ <p {a) + 



d (p (a) d- (p («) .i'- „ , .„ , , 



__ X + — — -;; + &c.: and, it we take the suc- 



da ■ da- \2 



cessive differentials of this equation according to a, we 



shall obtain a series of new equations, by means of which 



we can easily eliminate all the terms on the right hand side 



except the first, and the result will be <p (a) = <p (a + a?) — 



ling the terms to the other side we shall have f {a + x) = 

 X d. a(a + x) x"^ d'^ (a + x) 



From the form which we have here given to the develop- 

 ment, a number of consequences can be derived ; and 

 among these we shall first notice the theorem of Bernoulli. 



If we take a = o, then will a (x) = <S (o) + , - 



' 1 ax 



X* d^p (jf X' d^<p(x) , , r , s 



present the area of a curve, of which x and y are the •o- 



,. , , , , d0 X d"^ (S (x) dy 



ordmates,then ^ (x) ^ Sy ds; ~J- = y, ^^^ = ^ 



&c. and Sy dn = c (fl) + a'v -r^ + t^ —* 



^ I ^ >' r ^ l- 2 d X 1- 2- 3 d x'^ 



&c.; where the quantity (p (o) represents the constart, re 

 quisite for completing the integral. 



We shall next proceed to consider the form which the 

 binomial (a + x)" takes when subjected to this new spe- 

 cies of development. Here we have d. -, = 



, ^n -~ 1 d* (p {a + x) , , . n — 2. 



«• (a+x) ; ^,~ = n. n — 1. (a + x) i 



dx^ 



fcc, and thence we shall find (a -}- >r) = a + • r. 



w _ 1 n.7i— \ 2 , v" — 2 . 



(a + J-) ■ -- — -— — J ^ (a + 7?) + &c, 



