AND GENERA OE TERNARY QUADRATIC FORMS. 
295 
according as 12, A, is or is not divisible by any of the primes r ; i. e. according as 12, A, is 
not, or is prime to the greatest common divisor of 12 and A. In the expressions of R and 
R' the signs of summation extend to every combination of the equations 
({) = +!, or -1; (?)= + !, or -1; ({) = + l,or-l; (|) = + l,or-l; 
i e. the value of the continued product is to be determined on each of these suppositions, 
and the sum of these values is to be taken. From this definition it is evident that in 
the sum R, we may substitute for any factor of the form 
or 
»[■ + ?=?)>} 
a factor of the form 

or 
<»> 
outside the sign of summation. And similarly for any factor we may substitute 
the factor , 
outside the sign of summation, 
units, we obtain immediately 
Observing that the factors (54) and (55) are all positive 
R=n(i 
Again, if a prime r divide 12, or A,, the sum R' vanishes, being composed of pairs of 
terms equal in absolute magnitude and opposite in sign ; if, for example, r divide 12,, 
the two terms in one of which (^ji contained in is + 1 ’j anc ^ i n the other — 1, 
but which are in other respects identical, will. destroy one another. But if none of the 
primes r divide 12, or A,, we replace those factors of the general term of R', which 
mdccclxvii. 2 s 
