by a Table of Single Entry:- -^ -iM 433 



this term, such as •'•^ vfJKSo pr ;^i ,i9ri;t 



VI 92 qm ?m + l 9m + 2 • ' ^ q,i 



and 



•«<7...^.-«(/ „••• -«0.. +««,+««„+ •••+«« 



*9»,+ i "^ym+2 • • • "9n-r"qj-riig^-r . . . -r«j„, 

 one only is to be retained. The entire term is of course made 

 up exclusively of such pairs. 



Corollary. 

 If R(ai, flgj • • • ^n) denote any symmetrical algebraic function 

 whatever of a^, a^ . . . a^^, 

 



will contain cr, . «2 . cTg . . . «„ as a factor. In this formula Vj de- 

 notes the number of combinations of n things taken i and i 

 together. 



26 Lincoln's Inn Fields, 

 March 8, 1854. 



Postscript. 



In constructing a table of single entry for appplying the 

 formula 



4^b={a + bf-{a-bf, 

 i. e. 



ab=\{a + bf-\{a-.bf, 



it is only necessary to retain the integer part of the quarters of 

 the squares of all the numbers from 2 to the sum of the highest 

 of the values of a and b to which the application of the table is 



proposed to be restricted, because the fractional parts of I — ^ ] 

 /a—bY \ "« / 



and ( —^ ) will always destroy one another. A table for the 



multiplication of a ternary set of factors by means of the formula 

 al>c = ^{a + b + cf-~{a-\-b-c)^-—{a-b-^cf 



-^^{-a + b + cr, 



will imply the registration of the values of the 24th parts of all 

 numbers up to the liighest value of {a + b + c), and it becomes a 

 question of some practical interest to determine in what way the 

 fractional remainders of these 21th parts are to be dealt with. 



The formula last written may give rise to either of the two 

 Phil. May. S. 4. Vol. 7. No. 47. June 1854. 3 G 



