Section III, 1889 [ 15 ] Trans. Roy. Soc. Canada. 



V. — Cruees Matliemalicœ. 

 Ey Prof. N. F. Dupuis, Queeu's Uuiversity, Kingston. 



(Read May 8, 1889.) 

 I. 



An Elementary Method of obtaining the Discriminant of the 



General Quadratic. 



This is an elementary algebraic method ol' obtaining the discriminant of the general 

 quadratic in two variables. Algebraically the discriminant is defined as that function 

 of the coefficients whose vanishing denotes the i^ossibility of separating the quadratic 

 into two factors linear in x and y. 



Two well known methods of finding this function are given ; one in Hall and 

 Knight's " Higher Algebra," Art. 12*7, and both in Charles Smith's " Conies," Art. 3*7. The 

 present method, which I have never seen published, possesses advantages over those 

 mentioned inasmuch as it gives the forms of the factors into which the quadratic is 

 separable. 



Let 



ax^A- by--Y2hxy-\-2(jx+ 2fi/-\-c 



be the general quadratic. 

 Assume 



ax'+bf-+2hxij+2gx+2fy+c =(^ ax^-y+s ) Ç x+py-\-~y 



and equate coefficients, and we obtain 



1 



p = - (h+^h- — ab ), s = g+'/f — ac. 



a 



Then denoting 



>^A- — ab by H anS ■^g- — ac by G, 



the factors readily reduce to the form — 



- I ax+y(h-S)+g+G } | ax+y{h+ H)+g-G J . 

 Now since the factors do not contain/, we must have by equating coefficients of linear i/, 



2/= - (^+^ . h+S+ g=G . A=S) = 'U^gU-^GS). 



"Whence by substitution — 



{of — gKf = {(f — ac)Q^ — aV), 

 Or 



abc+2fgh—af—bg'—cK — 0. 



