by a Table of Single Entry. 435 



Kn =1^ when the said remainder is 7, 8, 9, 11, 13, 15, 16 or 17; 



and 



Kn =1 when the remainder is 19, 21 or 23 ; 



and let ol J + ^n be called the cubic respondent to N, and be 



denoted by R(N) ; 

 and let the exact value of — be called R'(N). 



Let 



W{a + b + c)-W{a + b-c)-W(a-b + c)-W{-a + b-\-c) 

 =Il(ffH-^i + c)— Il(«4-*— c)— R(ff— 6 + c)— R( — fl + 6 + e)+A. 



If in general we write R'(ra)— R(?i) = E(n), A must be of one 

 or the other of the two forms 



EK)-E(72,)-E(n3)-EK), 

 or 



EC«,)+EK)-E(»3)-EK), 



where n^, n^ n.^, n^ are supposed to be all positive integers. Now 



5 



it is easily seen that E(m) always lies mthin the limits +777; 



5 5 '^'* 



that is to say, it may reach up to ^ or down to —jr-., but can 



never transgress these values in either direction. Hence it is 

 obvious that A, which is made up of four terms, each of the form 

 E(w), can never be so great as +1 or so small as — 1, and con- 

 sequently A can only have one of the three values + 1, 0, — i. 



Hence, then, we may work with the tabular cubic respondents 

 in lieu of the exact cubic respondents ; if the result is an integer, 

 it is good without any correction ; if it is a fraction, i must be 

 added to, or taken away from it. And to ascertain which of 

 these processes is to be applied, it is only necessary to consider 

 whether the three factors to be multiplied are or not all of them 

 odd. 



In practically constructing a table of cubic respondents, it 

 would not be necessary actually to insert the fraction | in any 

 case; a dot over, or a stroke through the last integer, would 

 serve to denote that this fraction was to be understood. 



A table of quadratic respondents {i. e. of the integer parts of 

 the fourths of the square numbers) up to the base 20,000, has 

 been actually constructed and published by a M. Antoine Voisin, 

 under the title " Tables des Multiplications ou Logarithmes de 

 Nombres entiers depuis 1 jusqu^\ 20,000, au moyen desquelles 

 on pent multiplier tons les nombres qui n'excedent pas 20,000 

 par 20,000," &c. 12mo. :\ Paris, Firmin Didot, 1817. A copy 

 2 G2 



