280 Prof. Smith on CompleoG Binary Quadratic Forms, [June 16, 



is found by multiplying together the corresponding characters of the 

 factors ; so that, conversely, according as any character of a product of 

 two uneven factors is + 1 or —1, the two factors agree or differ in respect 

 of that character. 



The next Table assigns the supplementary characters proper to any 

 given determinant ; they depend on the residue of the determinant for the 

 modulus (l+^y. 



Table II. 



D = 



Characters. 



D = 



Characters. 



±(1+0.... 



/3 



+ 1 



7 



±(1-0.... 



a/3 



+ 2 



a 



±(3+0.... 



afty 



+ 3 



7 



±(3-0.... 





+ 3^• 



a 



+ 2 



a, y 



±(1-20 



y 



+ 2^■ 



y 



±(2 + 



ay 



2(1+0 .... 



a/3, y 



±(1 + 20 



y 



2(1-0 .... 





±(2-0 



ay 





a, y 











a, /3, y 







Of the eighteen propositions contained in this Table, it will suffice to 

 enunciate and demonstrate one. 



"If D = ±(3+0j niod (1+0^ and /is an uneven form of deter- 

 minant D, the uneven numbers represented by f, all have the character 

 a/3y= + 1, or else all have the character aPy= — 1." 



In the equation P^— DQ^=?wm', let us suppose that m and m' are un- 

 even ; then P is uneven because D is semieven ; also Q^ = ± 1, ±22, 4 or 0, 

 mod (1 + 0% according as the index of the highest power of l+i dividing 

 Q is 0, 1, 2, or > 2. If Q is uneven, mm'= + 3i or ±(2 + 0, mod 

 (1 + 0^ J if Q is semieven, mm' = +(l + 2i), mod (1+0^ ; if Q is even, 

 mm' = +ly mod (1+0^; «• ^. in all three cases mm' has the character 

 a/3y=l, and m and m' both have the character aj8y= + l, or else both 

 have the character a/3y= — 1. 



We add a third Table for the purpose of distinguishing between the 

 possible and impossible genera. In this Table is the greatest square 

 dividing D, P is uneven and primary*, I is the index of the highest power 

 of 1 + e dividing S, ^zr represents an uneven prime dividing P, o- an uneven 

 prime dividing S but not P. For brevity, the symbols w and <r are written 



instead of j^^J and pj. 



* By a primary uneven number we understand (with Lejeune Dirichlet) an uneven 

 number fi+n'i satisfying the congruences ii= I, mod 4, mod 2. 



