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LXX. Derivative Analysis i being a new and more comprehensive 

 Method of the Transformatio7i of Functions than anij hitherto 

 discovered : extending not only to the Extraction of the Roots 

 of Equations, but also to the Reductio7i of Qiiantitics from 

 the Multiples of Powers or Products to other equivalent Ex- 

 pressions, by which the Summation of any rational Series may 

 be readily effected. By Mr. Peter Nicholson. 

 5 Claremont-place, Judd-street. 



[Continued from p. 252.] 



Ex. 7. Til VIDE the series C+ — + - 



DIVIDE the series C+ — + ^' + ^^ + -^^ + &c. 

 V v" v3 t)l 



(which is the quotient of the preceding example 

 increased by the quantity C) by the binomial v—e (which is 

 the same divisor as in the two preceding examples). 



Divisor. 





C- 



v—e 

 Quotient 



+ ^+&c. 



&c. 

 By this operation we have the following derivative equa- 

 tions, Ba = eC 



C, = ^B3 + B, 

 D3 = eC, + C3 

 &c. 



Problem. 



To divide a quantity by a new divisor at each step equal to 

 that in the preceding step. 



Find the first part of the quotient and the remainder by 

 the first divisor. 



Find the second part of the quotient and the second re- 

 mainder by the second divisor. 



Proceed from step to step in this manner as far as may be 

 judged necessary, always sub'^tituting at each step as directed. 



The several equations being tabulated, will show the law by 



which 



