484 Mr. J. J. Sylvester on Multiplications, S^-c. 



subjoined cases, according as the numbers a, b, c correspond or 

 not to the lengths of a possible triangle, viz. 



U; «»'^-24^^» 24^^^ 24^^3 24^^4, 



or 



the quantities N,, Ng, Ng, N4 being all supposed to represent 

 positive integers. 



A very little consideration vidll show, that if we neglect frac- 

 tions in the table there may be entailed an error of 3, 1, 0, or 

 — 1 . Whether the error is, on the one hand, an error of an even 

 order (viz. or 2), or, on the other hand, of an odd order (viz. 

 1 or —1), would be at once obvious by looking to see whether the 

 formula, after neglecting the fractious, gave an odd result when 

 the result ought to be odd, and an even result when the result 

 ought to be even, or vice versa. And the nature of the result 

 as to whether it ouffht to be odd or even could be immediately 

 inferred from observing whether a, b, c were or were not all of 

 them odd numbers. But there would still remain an ambiguity 

 in the correction to be applied in either case, arising from the 

 doubt whether it should be zero or 2 in the one case, or whether 

 it should be + 1 or — 1 in the other case. 



This ambiguity might of course be removed by inserting in 



W 

 the table employed the first decimal place of ^, and increasing 



the decimal part in the final result to unity, or lowering it to 

 zero, according as its value might be greater or less than ^ ; 

 and it would be easy to ascertain the limits within which the 

 decimal digit in the I'esult must lie, and the range of values (of 

 which 5 is one) from which it is excluded. The same end may, 

 however, be gained much more elegantly and expeditiously, and 

 by a method more closely analogous to that employed for the 

 evolution of binary products, by the intervention of a very simple 

 expedient. 



The cubic residues in respect to the modulus 24 are easily 

 verified to be as follows : 0, 1, 3, 5, 7, 8, 9, 11, 13, 15, 16, 17, 



19, 21, 23. Let the tabular value of -r-7 be made jr— + Kn, 



t]>^3-] 24 L^-i J 



^ means the integer part of the quantity within the 



brackets, and Kn may have any one of the three values 0, i, 1, 

 viz. 



Kn=0 when the remainder of N^ to the divisor 24 is 0, 1, 

 3, or 5 ; 



