1820.] Demonstration of Taylor's Theorem, 8fc. 99 



Article III. 



Demonstration of Dr. Taylor's Theorem, with Examples. 

 By Mr. James Adams. 



(To Dr. Thomson.) 



SIR, Slomhouse, nctir Plymouth, Bee. il, 1819. 



Should you consider the following demonstration of 

 Dr. Taylor's theorem, together with the accompanying examples, 

 likely to benefit the young algebraist, your insertmg them iatht 

 Annals of Philosophy will much oblige, 



Your humble servant, 



James Adams. 



Prop. 1. — If (p represent a function composed of known and 

 unknown quantities, then will 



f, = (p + A ^+ — ^ — A- f + ~ • A' (p + &C. 



From the nature of increments 

 ? = <P 



?>,= <p +A(?5 = ip' 



<?>, = ?>/ =V +A<p' =^ + A^4-A^+ A"-^ = (p + 2A^-[-A"-(p = <p" 

 ^3=?/' =(p"+A^" =<p + 3A?5 + 3A-!p + A ? = p"' 

 f,=«'i"'='P'"+A<p'"=f + 4A^ + 6A'^^ + 4A^'^ + A^<f> 

 &c 



It is evident that the coefficients of <p, A (p, A- tp, A^ (p, &c. 

 are the same as the coefficients of the corresponding powers of 

 a binomial ; and, therefore, each in general is represented by 



A> "j z, > ^3 , ij^c. Hence the value of f^, as 



stated in the proposition. 



P;o;j. 2.— To find the nt\\ increment of the function ijs. 



A ip = (Bj — (p 



^''P = ?. -<?,-<?, + ?' = '?,- 2 ?>, + (p 



A' ^ = (P3 - 2?5„_ + ^^ - ,p^ + 2?>,-(p = s„-3?).^+ 3^, -?> 



A <p = tp^ - n <p,._, + ^ p„_, + ^y^ ?.._3 + &C. 



Prop. 3. — If (p represent a function composed of known and 

 unknown quantities, then will 



By prop. \, ^„ = „d^ + 11"^ .Z^ ^ + " ^""J.V"-'^ 



f/' ip + &c. Now since the increments d p, d'- p, (i* <p, tkc. are 

 always considered as " indefinitely or incomparably small," let 



G 2 



