126 Mr. J. J. Sylvester's Enumeration of the 



three first minors of U + W be zero, all are zero; we have 

 therefore to express that three quadi-atic functions of X have a 

 root in common. This implies the existence of two affirmative 

 conditions ; the contact of the two conies taken collectively may 

 therefore be still entitled of the second degree, although the 

 contact at each of the two points where it takes place is simple, 

 or of the first degree. 



These points are evidently defined by the equation 



and the ordinary algebraical solution of the equations U=0, 

 V = would natui-ally lead to the four systems 



x+ V~. y=0 z=0 

 x+ '/^.?/=0 z=0 



37-^/111.^ = Z = 



x—\/ — l.y=0 s=0 ; 

 the two tangents at the point of contact are a? + -v^ — 1?/=0, 

 X— \/ — ly = 0, and the coincident pair of lines containing the 

 intersections is z'^ssO. 



It may at first view appear strange, that whilst no condition 

 is required in order that U and V may be simultaneously meta- 

 moi-phosed into the forms of x^ + y'^ + z^, ax'^ + bz'^ + cz^, a, b 

 and c being all unequal, for this metamorphosis to be pos- 

 sible when any two become equal, not one but two conditions 

 must be satisfied. The i-eason of this is, that the coefficients of 

 transformation, which, as well as a, b, c, are functions of the 

 coefficients of the given quadratic functions, become infinite on 

 constituting between the said coefficients svich relations as are 

 necessary for satisfying the equation a = b, or a = c, or b = c, ex- 

 cept upon the assumption of some further particular relations 

 between them over and above that implied in such equality. 



In the ordinary case of diploidal contact, the first minors 

 having a factor in common, the complete determinant of U + \V, 

 this factor will enter twice into the complete determinant of 

 W + W; but it may enter three times : this will indicate, that 

 not only do the four intersections lie on a coincident pair of 

 lines, but furthermore, that there is but one pair of lines of any 

 kind on which they lie. 



for then P=0, Q=0 will imply R=0, S=() ; or, in like manner, by affirm- 

 ing any other two out of the foiu' necessary equations, and denying the other 

 equations. Observe, however, that all the reqiiii-ed equations may coexist in 

 the absence of such right of denial. 



