Theory of Canonical Forms and of Hyper determinants, 395 



:a3io!LiJ 'ui! 



h 



»WOf| 



ax-\-by hx + cy cx + dy 



bx + cy cx + dy dx-\-ey 



€X + dy dx-\-ey ex+fy 

 which dividing out each side of the equation by y^, immediately i 

 gives the identity required^ and the method is obviously general. 

 Turn we now to consider the mode of reducing a biquadratic 

 function of two letters to its canonical form^ videlicet 



(A +ffy)^ + (^^ + hY + ^^{fa +ffyY{hx + ky) . 



Let the given function be written ,^ ^^ 



ax"^ + 4ibaPy + Qcx^y^ 4- 4idxy^ + ey\ 



Let 



then we have 



k=h\2 mfh=fi A,j + X2=^j Xj.Xg=j 



f+h + 6fju=:a 



4/Xi + 4<h\ + 6//,(35i) - 4<b 



6/\,2 + eh\i + Qfi{si^ + 2*2) = 6c 



4f\^ + U\i -I- 6yLt(25iS2) = 4c? 



/Xi3 + ^X/ + 6/.522=e. 

 Eliminating / and h between the firsts second and third, the 

 second, third and fourth, and the third, fourth and fifth equa- 

 tions successively, we obtain 



(«- ^fi)Sc,- {b-Sfis^)s^ + (c-fiis^^ + 2s^)') =0 

 (b—Sfjbs^)s^— (c—fi{s^^+2s^)^Si + (d—Sfj,SiS^) =0. 

 (c— /a(5i« + 252)X+ {<^-'^H'SiSc^)si + {e-6fMSyS^=^,ij^ 



Let iiow 



and we shall have 



«52 ■- o^i + c — /A (8^2 — 4^1^) = 



bS2—' CS^ -f d—fJb{4SiS2 — 25i^) = 



€S^—ds^ + e—fjb{Ss^^--4!Si\) =0. 



Ui J 



re ^s 

 moil 



as^—bsi + (c + v) =0 

 Z>^2—(<J— 0^1 + ^=0 



{C -f v)52 — efej + 5 = 0. 



Hence v will be found from the cubic equation, 

 a; h; c + v 



c+v, d) e 



Tin 





