1864] Prof. Smith on Complex Binary Quadratic Forms, 293 

 pose that the uneven prime p is primary, it will suffice to show (i) that 

 if L^J = + 1' ^^en (-1) =1 ; (ii) that if (-1) =1, then 



j^-^J=l. (i) Let j^-^J = l; then, if w^— ^=0, mod p, w,- — 



is a form of determinant i; it therefore has the character a=l,i. e. 



(ii) Let (-1)*^^^"'^=!; then p=l, mod 4, and 

 the form (i, 0, ip) is an uneven form of determinant p ; it therefore has 



the character (—)= + 1 ; whence f— 1 = + 1 . 



(5) Similarly, if p=pQ-{-ip^ is an uneven and primary prime, to prove 



LZl = (- 1) « we shall show, (i) that if ^ =1, 



then (- 1) 8 = 1 ; (ii) that if (- 1) « =1, then [^J == 1 • 

 (i) Let J then there is a form of determinant 1+* and of 



the type w, ^ ~^ . this form has the character /3= + l ; there- 



fore (-1) 8 = + 1. (ii)Let(-l) » = + 1; then p is 

 either=l— 22, or=l, mod (1+2)'; if :p={\^iyk+\-2i, ([l+i]', i, 

 \—2ki) is an uneven form of determinant^ ; this form has the character 



7= + 1, and consequently it also has the character j^-^ J = + 1 ; there- 

 fore j^li2^ = [^(l±^'J== + l; if jp= (1+0^^+1, one or other of the 

 forms ([1 1> —k), and ([1 +2*]% 1 + [1 +*Ti 1—^) is an uneven form 

 of determinant py having the character j^-^ J = 1 J therefore in this case 



IV. The representation of Binary Forms of the principal Genus by 



Ternary Forms of Determinant 1 . 

 The solution of the general problem, " To find the representations (if 

 any) of a given binary by a given ternary quadratic form,'* depends, in the 

 case of complex as of real numbers, on the solution of the problem of equiva- 

 lence for ternary forms. Extending the methods of Gauss to the complex 

 theory, we find the necessary and sufficient condition for the primitive* 



* If a matrix of the type a , /3 



transforms a ternary into a binary quadratic form, the representation of the binary by 

 the ternary form is said to be primitive when the three determinants of the matrix are 

 relatively prime. 



z 2 



