372 On the solution of a System of* Equations. 



come under the general form which has just been solved. It 

 so happens, however, that in this particular case 



become respectively 



/l 



gl 



V 



/* 



gl 



h i 



fs 



g 3 



K 







1 



B" 



i - 



1 



"b 







i 

 c 



1 



"c 



l 

 B 







and the cubic equation is resolved without extraction of roots. 

 It follows from my theorem that the eight intersections of 

 three concentric surfaces of the second order can be found 

 by the solution of one cubic and one quadratic equation; and 

 in general, if we have </>, ^, 6 any three quadratic functions 

 of x 9 y y z 9 and (j) = 0, ^ = 0, = be the system of equations 

 to be solved, provided that we can by linear transformations 

 express <£, i|r, 6 under the form of 



U— aw 



V-bw 



W—cw, 



U, V, W being homogeneous functions, and w a non-homo- 

 geneous function of three new 7 variables, a?', y, z\ we can find 

 the eight points ot intersection of the three surfaces, of which 

 U, V, W are the characteristics, by the solution of one cubic 

 and one quadratic. But (as I am indebted to Mr. Cayley for 

 remarking to me) that this may be possible, implies the coin- 

 cidence of the vertices of one cone of each of the systems of 

 four cones in which the intersections of the three surfaces 

 taken two and two are contained. 



I may perhaps enter further hereafter into the discussion of 

 this elegant little theory. At present I shall only remark, 

 that a somewhat analogous mode of solution is applicable to 

 two equations, 



U=aP 2 



V = bP\ 



in which U, V are homogeneous quadratic functions, and P 

 some non-homogeneous function oix,y. 



We have only to make the determinant of fU +gV equal 



