446 WARING. 



Princijdes of translating Algebraic quantities into probable rela- 

 tions and annuities, (Ssc. By E. Waring, M.B. Lucasian Professor 

 of Mathematics at Cambridge, and Fellow of the Royal Societies 

 of London, Bononia and Gottingen, Caynbridge, Printed by J. Arch- 

 deacon, Printer to the University ; For J. Kicholson, Bookseller y in 

 Cambridge. 1792. 



This is an octavo pamphlet. Besides the leaf on which the 

 title is printed there are 59 pages of text, and then a page with 

 a few corrigenda. The work is excessively scarce ; for the use 

 of a copy I am indebted to the authorities of Queens' College, 

 Cambridge. 



829. The author and the printer seem to have combined their 

 efforts in order to render the work as obscure and repulsive as 

 possible ; and they have attained a fair measure of success. The 

 title is singularly inaccurate ; it is absurd to pretend to translate 

 algebraical quantities into probable relations or into annuities. 

 What Waring means is that algebraical identities may be trans- 

 lated so as to afford propositions in the Theory of Probabilities or 

 in the Theory of Annuities. 



830. Waring begins with a Lemma. He proposes to sum the 

 series 



1 + 2^-' r + 3^-^ r" + 4?-^r^ + S^'V* + . . . in infinitum. 



The sum will be 



A^-Br-\- Cr^ + Dr' + ... +y*-^ 



ii-ry 



The coefficients A, B, C ... are independent of r ; they must 

 be determined by multiplying up and equating coefficients. Thus 



B = 2''-''-z, 



G = 3*-^ - z2'-^ + ^ ^^~ -^^ , 



j^_,z-x ^ ^.-1 , g (^ - 1) 9.-1 z{z-l) (g-2) 

 U-^ -Z6 + ^ Z ^-^ . 



Proceeding in this way we shall find that in the numerator of 

 the fraction which represents the sum the last term is r""^ ; that 



